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• Sequences and Series • Convergence of Infinite Series • Tests of Convergence
• P-Series Test • Comparison Tests • Ratio test • Raabe’s test • Cauchy’s Root test • Integral test • Leibnitz’s test
• Absolute Convergence • Conditional convergence • Power series and Interval of convergence
• Summary of all Tests • Solved University Questions (JNTU) • Objective type of Questions
Chapter 1 Sequences and Series
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1.1 Sequence A function f:N → S, where S is any nonempty set is called a Sequence i.e., for each n∈ N, ∃ a unique element f(n) ∈ S. The sequence is written as f(1), f(2), f(3), ......f(n)...., and is denoted by {f(n)}, or <f(n)>, or (f(n)). If f(n) = na , the sequence is written as 1 2, ..... na a a and denoted by , { } ( ).< >n n na or a or a Here f(n) or na are the
thn terms of the Sequence.
Ex. 1. 1 , 4 , 9 , 16 ,......... 2n ,.....(or) 2n< >
1
23. .
.
.
.
.
.
.n
49
2n
N S
Ex. 2. 3 3 3 3 3
1 1 1 1 1, , ,..... ....( )1 2 3
⎛ ⎞⎜ ⎟⎝ ⎠
orn n
Ex. 3. 1, 1, 1......1..... or <1>
Sequences and Series
3
Ex 4: 1 , –1, 1, –1, ......... or ( ) 11 −− n
Note : 1. If S ⊆ R then the sequence is called a real sequence. 2. The range of a sequence is almost a countable set.
1.1.1 Kinds of Sequences
1. Finite Sequence: A sequence na< > in which 0 na n m N= ∀ > ∈ is said to be a finite Sequence. i.e., A finite Sequence has a finite number of terms.
2. Infinite Sequence: A sequence, which is not finite, is an infinite sequence.
1.1.2 Bounds of a Sequence and Bounded Sequence
1. If ∃ a number ‘M’ na∋ ≤ M, n∀ ∈ N, the Sequence na< > is said to be bounded above or bounded on the right.
Ex. 1 11, , ,2 3
,....... here 1 na n≤ ∀ ∈N
2. If ∃ a number ‘m’ ,na m n∋ ≥ ∀ ∈N, the sequence na< > is said to be bounded below or bounded on the left.
Ex. 1 , 2 , 3 ,.....here 1 na n≥ ∀ ∈N 3. A sequence which is bounded above and below is said to be bounded.
Ex. Let ( ) 11 1⎛ ⎞= − +⎜ ⎟⎝ ⎠
nna
n
n 1 2 3 4 ......
na -2 3/2 -4/3 5/4 ......
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From the above figure (see also table) it can be seen that m = –2 and M = 3
2.
∴ The sequence is bounded.
1.1.3 Limits of a Sequence
A Sequence na< > is said to tend to limit ‘l’ when, given any + ve number ' ',∈ however small, we can always find an integer ‘m’ such that ,− <∈ ∀ ≥na l n m , and we write nn
Lt a l→∞
= or →na l
Ex. If 2
2
12 3
+=
+nnan
then 12
< >→na .
1.1.4 Convergent, Divergent and Oscillatory Sequences 1. Convergent Sequence: A sequence which tends to a finite limit, say ‘l’ is
called a Convergent Sequence. We say that the sequence converges to ‘l’ 2. Divergent Sequence: A sequence which tends to ±∞ is said to be Divergent
(or is said to diverge). 3. Oscillatory Sequence: A sequence which neither converges nor diverges ,is
called an Oscillatory Sequence.
Ex. 1. Consider the sequence 2 , 3 4 5, , ,.....2 3 4
here 11= +nan
The sequence < >na is convergent and has the limit 1
1 11 1 1− = + − =nan n
and 1<∈
n whenever 1
>∈
n
Suppose we choose .001∈= , we have 1 .001<n
when n > 1000.
Ex. 2. If ( ) 13 1' '
= + − < >nn na a
n converges to 3.
Sequences and Series
5
Ex. 3. If ( )2 1 . ,= + − < >nn na n n a diverges.
Ex. 4. If ( )1 2 1 ,= + − < >nn na a
n oscillates between -2 and 2.
1.2 Infinite Series If nu< > is a sequence, then the expression 1 2 3 ........ .....nu u u u+ + + + + is called an
infinite series. It is denoted by 1
∞
=Σ nn
u or simply Σ nu
The sum of the first n terms of the series is denoted by ns i.e., 1 2 3 1 2 3...... ; , , ,....= + + + +n n ns u u u u s s s s are called partial sums.
1.2.1 Convergent, Divergent and Oscillatory Series
Let Σ nu be an infinite series. As ,→ ∞n there are three possibilities. (a) Convergent series: As ,→ ∞ →nn s a finite limit, say ‘s’ in which case the
series is said to be convergent and ‘s’ is called its sum to infinity.
Thus →∞
=nnLt s s (or) simply nLts s=
This is also written as 1 2 3 ..... ... .nu u u u to s+ + + + + ∞ = (or) 1
∞
=Σ =nn
u s (or)
simply .Σ =nu s (b) Divergent series: If → ∞ns or −∞ , the series said to be divergent. (c) Oscillatory Series: If ns does not tend to a unique limit either finite or infinite it
is said to be an Oscillatory Series.
Note: Divergent or Oscillatory series are sometimes called non convergent series.
1.2.2 Geometric Series
The series, 2 11 ..... ...−+ + + +nx x x is
(i) Convergent when 1<x , and its sum is 11− x
(ii) Divergent when 1≥x .
(iii) Oscillates finitely when x = -1 and oscillates infinitely when x < -1.
Proof: The given series is a geometric series with common ratio ‘x’
∴ 11−
=−
n
nxsx
when 1≠x [By actual division – verify]
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(i) When 1:<x
1 11 1 1→∞ →∞ →∞
⎛ ⎞⎛ ⎞= − =⎜ ⎟⎜ ⎟− − −⎝ ⎠ ⎝ ⎠
n
nn n n
xLt s Lt Ltx x x
since 0 as ⎡ ⎤→ → ∞⎣ ⎦nx n
∴ The series converges to 11− x
(ii) When 11:1
−≥ =
−
n
nxx sx
and → ∞ns as → ∞n
∴ The series is divergent. (iii) When x = –1: when n is even, 0→ns and when n is odd, 1→ns ∴ The series oscillates finitely. (iv) When 1,< − → ∞nx s or −∞ according as n is odd or even. ∴ The series oscillates infinitely.
1.2.3 Some Elementary Properties of Infinite Series 1. The convergence or divergence of an infinites series is unaltered by an addition or
deletion of a finite number of terms from it. 2. If some or all the terms of a convergent series of positive terms change their signs,
the series will still be convergent. 3. Let Σ nu converge to ‘s’ Let ‘k’ be a non – zero fixed number. Then Σ nku converges to ks. Also, if Σ nu diverges or oscillates, so does Σ nku
4. Let Σ nu converge to ‘l’ and Σ nv converge to ‘m’. Then (i) ( )Σ +n nu v converges to ( l + m ) and (ii) ( )Σ +n nu v converges to ( l – m )
1.2.4 Series of Positive Terms Consider the series in which all terms beginning from a particular term are +ve.
Let the first term from which all terms are +ve be 1u . Let Σ nu be such a convergent series of +ve terms. Then, we observe that the convergence is unaltered by any rearrangement of the terms of the series.
1.2.5 Theorem
If Σ nu is convergent, then 0→∞
=nnLt u .
Proof : 1 2 ......= + + +n ns u u u 1 1 2 1,......− −= + + +n ns u u u , so that, 1−= −n n nu s s
Sequences and Series
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Suppose Σ =nu l then →∞
=nnLt s l and 1−→∞
=nnLt s l
∴ ( )1−→∞ →∞= −n n nn n
Lt u Lt s s ; 1 0−→∞ →∞− = − =n nn n
Lt s Lt s l l
Note: The converse of the above theorem need not be always true. This can be observed from the following examples.
(i) Consider the series, 1 1 11 ....... ....2 3
+ + + + +n
; 1 , 0→∞
= =n nnu Lt u
n
But from p-series test (1.3.1) it is clear that 1
Σn
is divergent.
(ii) Consider the series, 2 2 2 2
1 1 1 1..... ......1 2 3
+ + + + +n
2
1 , 0,→∞
= =n nnu Lt u
n by p series test, clearly 2
1Σ
n converges,
Note : If 0→∞
≠nnLt u the series is divergent;
Ex. 2 12
−=
n
n nu , here 1→∞
=nnLt u ∴ Σ nu is divergent.
1.3 Tests for the Convergence of an Infinite Series In order to study the nature of any given infinite series of +ve terms regarding convergence or otherwise, a few tests are given below.
1.3.1 P-Series Test
The infinite series, 1
1 1 1 1 ......,1 2 3
∞
=Σ = + + +p p p pn n
is
(i) Convergent when p > 1, and (ii) Divergent when 1p ≤ . (JNTU 2002, 2003)
Proof :
Case (i) Let p > 1; 1,3 2> >p pp ; ⇒ 1 13 2
<p p
∴ 1 1 1 1 22 3 2 2 2
+ < + =p p p p p
Similarly, 1 1 1 1 1 1 1 1 44 5 6 7 4 4 4 4 4
+ + + < + + + =p p p p p p p p p
1 1 1 8.... ,8 9 16 8
+ + + <p p p p and so on.
Adding we get
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1 2 4 81 ....
2 4 8Σ < + + + +p p p pn
i.e., ( ) ( ) ( )1 2 1 3 1
1 1 1 11 ......2 2 2− − −
Σ < + + + +p p p pn
The RHS of the above inequality is an infinite geometric series with common
ratio ( )1
1 1 since 12 − < >p p The sum of this geometric series is finite.
Hence 1
1∞
=Σ pn n
is also finite.
∴ The given series is convergent.
Case (ii) Let p =1; 1 1 1 11 ......2 3 4
Σ = + + + +pn
We have, 1 1 1 1 13 4 4 4 2
+ > + =
1 1 1 1 1 1 1 1 15 6 7 8 8 8 8 8 2
+ + + > + + + =
1 1 1 1 1 1 1....... .....9 10 16 16 16 16 2
+ + > + + = and so on
∴ 1 1 1 1 1 1 11 .....2 3 4 5 6 7
⎛ ⎞ ⎛ ⎞Σ = + + + + + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠pn
1 1 11 .....2 2 2
≥ + + + +
The sum of RHS series is ∞
1 1since 12 2 →∞
− +⎛ ⎞= + = = ∞⎜ ⎟⎝ ⎠
n nn
n ns and Lt s
∴ The sum of the given series is also ∞ ; ∴ 1
1∞
=Σ pn n
( p = 1 ) diverges.
Case (iii) Let p<1, 1 1 11 .....2 3
Σ = + + +p p pn
Since , ,
1 1 1 11, ......2 2 3 3
< > >p pp and so on
∴ 1 1 1 11 .......2 3 4
Σ > + + + +pn
From the Case (ii), it follows that the series on the RHS of above inequality is divergent.
Sequences and Series
9
∴ 1
Σ pn is divergent , when P < 1
Note: This theorem is often helpful in discussing the nature of a given infinite series.
1.3.2 Comparison Tests 1. Let Σ nu and Σ nv be two series of +ve terms and let Σ nv be convergent. Then Σ nu converges, (a) If ,≤ ∀n nu v n N∈
(b) or ≤ ∀ ∈n
n
u k n Nv
where k is > 0 and finite.
(c) or →n
n
uv
a finite limit > 0
Proof : (a) Let Σ =nv l (finite) Then, 1 2 1 2..... ...... ..... ..... 0+ + + + ≤ + + + ≤ >n nu u u v v v l Since l is finite it follows that Σ nu is convergent
(c) ,≤ ⇒ ≤ ∀ ∈nn n
n
u k u kv n Nv
, since Σ nv is convergent and k (>0) is finite,
Σ nkv is convergent ∴ Σ nu is convergent.
(d) Since →∞
n
nn
uLtv
is finite, we can find a +ve constant ,∋ < ∀ ∈n
n
uk k n Nv
∴ from (2) , it follows that Σ nu is convergent 2. Let Σ nu and Σ nv be two series of +ve terms and let Σ nv be divergent. Then
Σ nu diverges, * 1. If ,≥ ∀ ∈n nu v n N
or * 2. If ,≥ ∀ ∈n
n
u k n Nv
where k is finite and 0≠
or * 3. If →∞
n
nn
uLtv
is finite and non-zero.
Proof: 1. Let M be a +ve integer however large it may be. Science Σ nv is divergent, a
number m can be found such that 1 2 ..... ,+ + + > ∀ >nv v v M n m ∴ ( )1 2 ..... ,+ + + > ∀ > ≥n n nu u u M n m u v ∴ Σ nu is divergent
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2. 1 ≥ ∀nu kv n Σ nv is divergent ⇒ Σ nkv is divergent ∴ Σ nu is divergent
3. Since →∞
n
nn
uLtv
is finite, a + ve constant k can be found such that ,> ∀n
n
u k nv
(probably except for a finite number of terms ) ∴ From (2), it follows that Σ nu is divergent. Note :
(a) In (1) and (2), it is sufficient that the conditions with * hold ∀ > ∈n m N Alternate form of comparison tests : The above two types of comparison tests 2.8.(1) and 2.8.(2) can be clubbed together and stated as follows :
If Σ nu and Σ nv are two series of + ve terms such that n
nn
uLtv→∞
= k, where k is
non- zero and finite, then Σ nu and Σ nv both converge or both diverge.
(b) 1. The above form of comparison tests is mostly used in solving problems. 2. In order to apply the test in problems, we require a certain series Σ nv whose
nature is already known i.e., we must know whether Σ nv is convergent are divergent. For this reason, we call Σ nv as an ‘auxiliary series’.
3. In problems, the geometric series (1.2.2.) and the p-series (1.3.1) can be conveniently used as ‘auxiliary series’.
Solved Examples
EXAMPLE 1 Test the convergence of the following series:
(a) 3 4 5 6 .....1 8 27 64
+ + + + (b) 4 5 6 7 .....1 4 9 16
+ + + + (c) ( )1 44
11
∞
=
⎡ ⎤Σ + −⎢ ⎥⎣ ⎦nn n
SOLUTION
(a) Step 1: To find " "nu the thn term of the given series. The numerators 3, 4, 5, 6......of the terms, are in AP.
thn term ( )3 1 .1 2= + − = +nt n n
Denominators are 3 3 3 31 ,2 ,3 ,4 ..... thn term = 3n ; ∴ 3
2+=n
nun
Step 2: To choose the auxiliary series Σ nv . In ,nu the highest degree of n in the numerator is 1 and that of denominator is 3.
Sequences and Series
11
∴ we take, 3 1 2
1 1−= =nv
n n
Step 3: 23
2 2 21 1,→∞ →∞ →∞ →∞
+ + ⎛ ⎞= × = = + =⎜ ⎟⎝ ⎠
n
n n n nn
u n nLt Lt n Lt Ltv n n n
which is non- zero and
finite.
Step 4: Conclusion: 1→∞
=n
nn
uLtv
∴ Σ nu and Σ nv both converge or diverge (by comparison test). But 2
1Σ = Σnv
n is
convergent by p-series test (p = 2 > 1); ∴ Σ nu is convergent.
(b) 4 5 6 7 .....1 4 9 16
+ + + +
Step 1: 4 , 5, 6, 7, .....in AP , ( )4 1 1 3= + − = +nt n n ∴ 2
3+=n
nun
Step 2: Let 1Σ =nv
n be the auxiliary series
Step 3: 2
3 31 1→∞ →∞ →∞
+⎛ ⎞ ⎛ ⎞= × = + =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
n
n n nn
u nLt Lt n Ltv n n
, which is non-zero and finite.
Step 4: ∴ By comparison test, both Σ nu and Σ nv converge are diverge together.
But 1Σ = Σnv
n is divergent, by p-series test (p = 1); ∴ Σ nu is divergent.
(c) ( )1 44
11
∞
=
⎡ ⎤Σ + −⎢ ⎥⎣ ⎦nn n
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⎟⎠⎞
⎜⎝⎛ +=
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ += 1–11–11
41
4
41
44
nnn
nn
4 8 4 8
1 1 11 1 1 34 41 . ..... 1 .....
4 2! 4 32
⎡ ⎤⎛ ⎞−⎜ ⎟⎢ ⎥ ⎡ ⎤⎝ ⎠⎢ ⎥= + + + − = − +⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦
n nn n n n
3 7 3 4
1 3 1 1 3.... .....4 32 4 32
⎡ ⎤= − + = − +⎢ ⎥⎣ ⎦n n n n
Here it will be convenient if we take 3
1=nv
n
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4
1 1 1..... ,4 32 4→∞ →∞
⎛ ⎞= − + =⎜ ⎟⎝ ⎠
n
n nn
uLt Ltv n
which is non-zero and finite
∴ By comparison test, Σ nu and Σ nv both converge or both diverge. But by p-
series test 3
1Σ =nv
n is convergent. (p = 3 > 1); ∴ Σ nu is convergent.
EXAMPLE 2
If 3 2
34
3 12 3 5
+=
+ +n
nun n
show that Σ nu is divergent
SOLUTION
As n increases, nu approximates to
1 2 13 2 3 3 3
1 3 1 134 4 4 124
3 3 3 1.2 2 2
= × =n nn nn
∴ If we take 112
1=nv
n ,
13
14
3
2→∞=n
nn
uLtv
which is finite.
[(or) Hint: Take 1 2
1−=n l lv
n , where 1l and 2l are indices of ‘n’ of the largest terms
in denominator and nominator respectively of nu . Here 3 2 14 3 12
1 1−
= =nvnn
]
By comparison test, Σ nv and Σ nu converge or diverge together. But 112
1Σ = Σnv
n is
divergent by p – series test ( 1since 112
= <p )
∴ Σ nu is divergent.
EXAMPLE 3
Test for the convergence of the series. 1 2 3 4 ......2 3 4 5
+ + + +
SOLUTION
Here, 1
=+nnu
n; Take 1 1 0
2 2
1 1 1−
= = =nvn
n, 1 111
→∞ →∞= =
+
n
n nn
uLt Ltv
n
(finite)
Sequences and Series
13
Σ nv is divergent by p – series test. (p = 0 < 1) ∴ By comparison test, Σ nu is divergent, (Students are advised to follow the procedure given in ex. 1.2.9(a) and (b) to find “ nu ” of the given series.)
EXAMPLE 4
Show that 1 1 11 ....... .....1 2
+ + + + +n
is convergent.
SOLUTION
1=nu
n (neglecting 1st term )
= 1
1 1 11.2.3...... (2 )1.2.2.2..... 1 −< =
− nn n times
∴ 2 3
1 1 11 ......2 2 2
Σ < + + + +nu
which is an infinite geometric series with common ratio 1 12
<
∴ 1
12 −Σ n is convergent. (1.2.3(a)). Hence Σ nu is convergent.
EXAMPLE 5
Test for the convergence of the series, 1 1 1 .......1.2.3 2.3.4 3.4.5
+ + +
SOLUTION
( )( )1
1 2=
+ +nun n n
; Take 3
1=nv
n
3
31
1 21 1→∞ →∞
= =⎛ ⎞⎛ ⎞+ +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
n
n nn
u nLt Ltv n
n n
(finite)
∴ By comparison test, Σ nu , and Σ nv converge or diverge together. But by p-series test,
Σ nv = 3
1Σ
n is convergent ( p = 3 > 1 ); ∴ Σ nu is convergent .
EXAMPLE 6
If 4 41 1,= + − −nu n n show that Σ nu is convergent. [JNTU, 2005]
SOLUTION
1 12 22 2
4 4
1 11 1⎛ ⎞ ⎛ ⎞= + − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
nu n nn n
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24 8 12 4 8 12
1 1 1 1 1 11 .... 1 .....2 8 16 2 8 16
⎡ ⎤⎛ ⎞ ⎛ ⎞= + − + − − − − − −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦n
n n n n n n
24 12 2 10
1 1 1 1.... 1 ....8 8
⎡ ⎤ ⎡ ⎤= + + = + +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦n
n n n n
Take 2
1=nv
n , hence 1
→∞=n
nn
uLtv
∴ By comparison test, Σ nu and Σ nv converge or diverge together. But 2
1Σ =nv
n is
convergent by p –series test (p = 2 > 1) ∴ Σ nu is convergent.
EXAMPLE 7
Test the series 1 1 1 .....1 2 3
+ + ++ + +x x x
for convergence.
SOLUTION
1=
+nun x
; take 1=nv
n, then 1
1= =
+ +
n
n
u nxv n xn
1 11;1
→∞
⎛ ⎞⎜ ⎟
= Σ = Σ⎜ ⎟⎜ ⎟+⎝ ⎠
nnLt vx n
n
is divergent by p-series test (p =1 )
∴ By comparison test, Σ nu is divergent.
EXAMPLE 8
Show that 1
1sin∞
=
⎛ ⎞Σ ⎜ ⎟⎝ ⎠n n
is divergent.
SOLUTION
1sin ;⎛ ⎞= ⎜ ⎟⎝ ⎠
nun
take 1=nv
n
0
1sinsin
1→∞ →∞ →
⎛ ⎞⎜ ⎟⎝ ⎠= =
⎛ ⎞⎜ ⎟⎝ ⎠
n
n n tn
u tnLt Lt Ltv t
n
(where 1 )=t n 1=
∴ ,Σ Σn nu v both converge or diverge . But
1Σ = Σnv
n is divergent
( p -series test, 1p = ); ∴ Σ nu is divergent.
Sequences and Series
15
EXAMPLE 9
Test the series 1 1sin− ⎛ ⎞Σ ⎜ ⎟⎝ ⎠n
for convergence.
SOLUTION
1 1sin−=nun
; Take 1=nv
n
11
sin1
⎛ ⎞− ⎜ ⎟⎝ ⎠
→∞ →∞=
⎛ ⎞⎜ ⎟⎝ ⎠
nn
n nn
uLt Ltv
n
; = 1
0
11 sinsin
−
→
⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
Lt Takingnθ
θ θθ
But Σ nv is divergent. Hence Σ nu is divergent.
EXAMPLE 10
Show that the series 2 3
2 3 3
1 2 31 .....2 3 4
+ + + + is divergent.
SOLUTION
Neglecting the first term, the series is .....43
32
21
4
3
3
2
2 +++ . Therefore
( ) ( )( )1 1 11 1 11 . 1
+= = =+ ++ ⎛ ⎞ ⎛ ⎞+ +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
n n n
n n nn
n n nu nn nn n n
n n
= 11 11 1⎛ ⎞⎛ ⎞+ +⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
n
nn n
;
Take 1=nv
n
∴ 1 1 11 1 11 1 1 .1
→∞ →∞ →∞= = =
⎛ ⎞⎛ ⎞ ⎛ ⎞+ + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
nn nn n n
n
uLt Lt Ltv e
n n n
which is finite and 1Σ = Σnv
n is divergent by p –series test ( p = 1)
∴ Σ nu is divergent.
EXAMPLE 11
Show that the series 1 3 5 .......
1.2.3 2.3.4 3.4.5+ + + ∞ is convergent. (JNTU 2000)
SOLUTION
1 3 5 .......
1.2.3 2.3.4 3.4.5+ + + ∞
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thn term = ( )( ) 2
122 1 1 .
1 21 2 1 1n
n nun n n n
n n
⎛ ⎞−⎜ ⎟− ⎝ ⎠= =+ + ⎛ ⎞⎛ ⎞+ +⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
Take 2
1nv
n=
( )( )2 2
121 1
1 21 1n
n nn
u nLt Ltv n n
n n→∞ →∞
⎛ ⎞−⎜ ⎟ ⎛ ⎞⎝ ⎠= ÷ ⎜ ⎟⎝ ⎠+ +
( )( )
2 0 21 0 1 0
n
nn
uLtv→∞
−= =
+ + which is finite and non-zero
∴ By comparison test nu∑ and nv∑ converge or diverge together
But nv∑ = 2
1∑ n is convergent. ∴ nu∑ is also convergent.
EXAMPLE 12
Test whether the series 1
11n n n
∞
= + +∑ is convergent (JNTU 1997, 1999, 2003)
SOLUTION
The given series is 1
11n n n
∞
= + +∑
1
1nun n
=+ +
=( )( )
1 11 1
n n n nn n n n
+ −= + −
+ + + −
1
2
2
1 1 11 1 1 ..... 12 8nu n n
n n n
⎧ ⎫ ⎧ ⎫⎪ ⎪⎛ ⎞ ⎛ ⎞= + − = + − + −⎨ ⎬ ⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎩ ⎭⎪ ⎪⎩ ⎭
⎭⎬⎫
⎩⎨⎧ +=
⎭⎬⎫
⎩⎨⎧ += ....
81–
211...
81–
21
2 nnnnnun
Sequences and Series
17
Take 1
nvn
=
1 1 2 1 1......
2 8 2n
n nn
uLt Ltv nn n→∞ →∞
⎛ ⎞⎧ ⎫= − + ÷ =⎨ ⎬ ⎜ ⎟⎩ ⎭ ⎝ ⎠
which is finite and non-zero . Using comparison test nu∑ and nv∑ converge or diverge together.
But nv∑ = 1n∑ is divergent ( )1since 2p =
∴ nu∑ is also divergent.
EXAMPLE 13
Test for convergence 3 3
1
1n
n n∞
=
⎡ ⎤+ −⎣ ⎦∑ [JNTU 1996, 2003, 2003]
thn term ( )1
3
3 3 6
1 1 11 1 13 31 1 1 . ..... 13 1.2nu n n
n n n
⎡ ⎤−⎡ ⎤⎛ ⎞ ⎢ ⎥⎢ ⎥= + − = + + + −⎜ ⎟ ⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦ ⎣ ⎦
= 2 5 2 3
1 1 1 1 1...... ......3 9 3 9n n n n
⎛ ⎞− + = − +⎜ ⎟⎝ ⎠
; Let 2
1nv
n=
Then ( )31 1 1.... 03 39
nn nn
uLt Ltv n→∞ →∞= − + = ≠
∴ By comparison test, nu∑ and nv∑ both converge or diverge.
But nv∑ is convergent by p -series test ( )since 2 1p = > ∴ nu∑ is convergent.
EXAMPLE 14
Show that the series, 2 3 4 .......
1 2 3p p p+ + + is convergent for p > 2 and divergent for
2p ≤
SOLUTION
thn term of the given series = ( ) ( )
1
1 11 11n p p p
nn n nun n n −
+ ++= = =
Let us take 1
1 ; 1 0;nn p n n
uv Lt vn − →∞= = ≠
Engineering Mathematics - I
18
∴ nu∑ and nv∑ both converge or diverge by comparison test.
But nv∑ = 11
pn −∑ converges when p -1>1 ; i.e., p >2 and diverges when
1 1p − ≤ i.e 2p ≤ ; Hence the result.
EXAMPLE 15
Test for convergence 1
2
1
2 33 1
n
nn
∞
=
⎛ ⎞+⎜ ⎟+⎝ ⎠
∑ (JNTU 2003)
SOLUTION
( )( )
1232 1 2
13 1 3
nn
nn
n
u⎡ ⎤+⎢ ⎥= ⎢ ⎥
+⎢ ⎥⎣ ⎦
; Take 23
n
n nv = ;
1231 2
11 3
nn
n n
uv
⎛ ⎞+⎜ ⎟=⎜ ⎟+⎝ ⎠
1 0n
nn
uLtv→∞
= ≠ ; ∴ By comparison test, nu∑ and nv∑ behave the same way.
But nv∑ =3
2 2
1
2 2 2 2 .....,3 3 3 3
n
n
∞
=
⎛ ⎞ ⎛ ⎞= + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
∑ which is a geometric series with
common ratio 23 (<1) ∴ nv∑ is convergent. Hence nu∑ is convergent.
EXAMPLE 16
Test for convergence of the series, 1 4 9 ......
4.7.10 7.10.13 10.13.16+ + + (JNTU 2003)
SOLUTION
4, 7, 10,..............is an A . P; ( )4 1 3 3 1nt n n= + − = +
7, 10, 13,............is an A . P; ( )7 1 3 3 4nt n n= + − = +
and 10 , 13 , 16 ,.............is an A. P; ( )10 1 3 3 7nt n n= + − = +
∴ ( )( )( ) ( ) ( ) ( )
2 2
71 43 1 3 4 3 7 3 1 .3 1 .3 13 3 3n
n nun n n n n nn n n
= =+ + + + + +
=( )( )( )
171 427 1 1 13 3 3n n n n+ + +
;
Sequences and Series
19
Taking 1
nvn
= , we get
1 027
n
nn
uLtv→∞
= ≠ ; ∴ By comparison test, both nu∑ and nv∑ behave in the same
manner. But by p –series test, nv∑ is divergent, since p = 1. ∴ nu∑ is divergent .
EXAMPLE 17
Test for convergence 2
3 2
2 5 14 7 2
n nn n
− +− +∑ (JNTU 2003)
SOLUTION
thn term of the given series = 2
3 2
2 5 14 7 2n
n nun n
− +=
− +
Let 2
1nv
n=
( )
22
33
5 12.
17 24n
n nn
nu nn nLt Ltv n n n
→∞ →∞
⎡ ⎤− +⎢ ⎥= ×⎢ ⎥− +⎢ ⎥⎣ ⎦
= ( )
2
3
5 12 2 047 24n
n nLtn n
→∞
⎡ ⎤− +⎢ ⎥ = ≠⎢ ⎥− +⎢ ⎥⎣ ⎦
∴ By comparison test, nu∑ and nv∑ both converge or diverge.
But nv∑ is convergent. [p series test – p = 2 > 1] ∴ nu∑ is convergent .
EXAMPLE 18
Test the series nu∑ , whose thn term is ( )2
14n i−
SOLUTION
( )2
14nu
n i=
−; Let 2
1nv
n= ,
( )2
22
1 044
n
n nn
u nLt Ltv in n
→∞ →∞
⎡ ⎤⎢ ⎥= = ≠⎢ ⎥
−⎢ ⎥⎣ ⎦
∴ nu∑ and nv∑ both converge or diverge by comparison test. But nv∑ is
convergent by p –series test ( )2 1p = > ; ∴ nu∑ is convergent.
Note: Test the series ∑∞
=12 141
n –n
Engineering Mathematics - I
20
EXAMPLE 19
If 1 1.sinnun n
⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
, show that nu∑ is convergent .
SOLUTION
Let 2
1nv
n= , so that nv∑ is convergent by p –series test .
( )
( ) 0
1sin sin1
n
n n tn
u tnLt Lt Ltv t
n→∞ →∞ →
⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
where t = 1/n, Thus 1 0n
nn
uLtv→∞
⎛ ⎞= ≠⎜ ⎟
⎝ ⎠
∴ By comparison test, nu∑ is convergent.
EXAMPLE 20
Test for convergence 1 1tan( )nn∑
SOLUTION
Take 32
1nv
n= ; 1 0n
n n
uLt v→∞
⎡ ⎤ = ≠⎢ ⎥⎣ ⎦ ( as in above example)
Hence by comparison test, nu∑ converges as nv∑ converges.
EXAMPLE 21
Show that 2
1
1sinn n
∞
=
⎛ ⎞⎜ ⎟⎝ ⎠
∑ is convergent.
SOLUTION
Let nu 2 1sinn
⎛ ⎞= ⎜ ⎟⎝ ⎠
; Take 2
1nv
n= ,
( ) 22
0
1sin sin1
n
n n tn
u tnLt Lt Ltv tn
→∞ →∞ →
⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥= =⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎝ ⎠⎝ ⎠ ⎣ ⎦
where 1t n= ; 21 1 0nn n
uLt v→∞
⎛ ⎞ = = ≠⎜ ⎟⎝ ⎠
∴ By comparison test, nu∑ and nv∑ behave the same way.
But nv∑ is convergent by p- series test, since p = 2 > 1; ∴ nu∑ is convergent.
Sequences and Series
21
EXAMPLE 22
Show that ( )2
1log n
n n
∞
=∑ is divergent.
SOLUTION 1
lognu n n= ; 1 1log 2 1 2log 2 2 2log 2 2< ⇒ < ⇒ > ;
Similarly 1 1 1 1..... ,3log 3 3, log n Nn n n> > ∈
∴ 1 1logn n n∑ > ∑ ; But 1
n∑ is divergent by p-series test.
By comparison test, given series is divergent. [If nv∑ is divergent and n nu v n≥ ∀ then
nu∑ is divergent.] (Note : This problem can also be done using Cauchy’s integral Test.
EXAMPLE 23
Test the convergence of the series ( ) ( )1
r s
nc n d n
∞− −
=
+ +∑ , where c, d, r, s are all +ve.
SOLUTION
The thn term of the series ( ) ( )
1n r su
c n d n= =
+ +.
Let 1
n r svn += Then
1
1 . 1 1 1
r sn
r s r sn r s
u nv c d c dn n
n n n n
+
= =⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
1 0n
nn
uLtv→∞
= ≠ , ∴ nu∑ and nv∑ both converge are diverge, by comparison test.
But by p-series test, nv∑ converges if (r + s) > 1 and diverges if (r + s) ≤ 1
∴ nu∑ converges if ( r + s ) > 1 and diverges if ( r + s ) ≤ 1.
EXAMPLE 24
Show that ( )11
1
nn∞
− +∑ is divergent.
SOLUTION
( )11
11
.n
nn
u nn n
− += = Take
1nv
n= ; 1
1 1 0n
n n nn
uLt Ltv n→∞ →∞
= = ≠
Engineering Mathematics - I
22
For let 11
n nLt y
n→∞= say; log y =
1 .lognLt n
n→∞− =
10
1n
nLt→∞
− =
∴ 0 1y e= = (∞⎛ ⎞
⎜ ⎟∞⎝ ⎠ using L Hospitals rule)
By comparison test both nu∑ and nv∑ converge or diverge. But p-series test,
nv∑ diverges (since p =1); Hence nu∑ diverges.
EXAMPLE 25
Test for convergence the series ( )
( ) ( )1
r
p qn
n an b n c
∞
=
+
+ +∑ , a, b, c , p, q, r, being +ve.
SOLUTION
( )( ) ( )
( )( ) ( )qqpp
rr
qp
r
n
ncnn
bnn
an
cnbnanu
++
+=
+++
=11
1 =
( )( ) ( )
11 .1 1
r
p qp q r
an
n b cn n
+ −
+
+ +;
Take 1
n p q rvn + −= ; 1 0n
nn
uLtv→∞
= ≠ ;
Applying comparison tests both nu∑ and nv∑ converge or diverge.
But by p-series test, nv∑ converges if ( p+ q –r ) > 1and diverges if ( p + q –r) ≤ 1.
Hence nu∑ converges if ( p + q – r ) > 1 and diverges if ( p + q – r) ≤ 1.
EXAMPLE 26 Test the convergence of the following series whose thn terms are:
(a) ( )
( )( )( )3 4
2 1 2 3 2 5n
n n n+
+ + + ; (b)
1tann
; (c) 2
1 13
nnn n
+⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟+⎝ ⎠⎝ ⎠
(d) ( )
13 5n n+
; (e) 1.3nn
SOLUTION
(a) Hint : Take 2
1 ;n nv vn
= ∑ is convergent; 3 08
n
nn
uLtv→∞
⎛ ⎞= ≠⎜ ⎟
⎝ ⎠ (Verify)
Apply comparison test: nu∑ is convergent [the student is advised to work out this problem fully ]
Sequences and Series
23
(b) Proceed as in Example 8; nu∑ is convergent.
(c) Hint : Take 2
1nv
n= ;
( )
( )1
3 2
1 1 031
nnn
nn nn
u eLt Ltv e e
n→∞ →∞
+⎛ ⎞= = = ≠⎜ ⎟
⎝ ⎠ +
2
1nv
n= is convergent (work out completely for yourself )
(d) 1 1 1.
3 5 5 315
n n n n nu = =
+ ⎡ ⎤⎛ ⎞+⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦
; Take 15n nv = ; 1 0n
nn
uLtv→∞
⎛ ⎞= ≠⎜ ⎟
⎝ ⎠
nu∑ and nv∑ behave the same way. But nv∑ is convergent since it is a
geometric series with common ratio 1 15
<
∴ nu∑ is convergent by comparison test .
(e) 1 1 ,.3 3n n n N
n≤ ∀ ∈ , since .3 3n nn ≥ ;
∴ 1 1.3 3n nn
≤∑ ∑ …..(1)
The series on the R.H .S of (1) is convergent since it is geometric series with 1 13
r = < .
∴ By comparison test 1.3nn∑ is convergent.
EXAMPLE 27
Test the convergence of the following series.
(a) 2 2 2 2 2 2 2 2 2
1 2 1 2 3 1 2 3 411 2 1 2 3 1 2 3 4
+ + + + + ++ + +
+ + + + + + + ...............
(b) 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3
1 2 1 2 3 1 2 3 411 2 1 2 3 1 2 3 4
+ + + + + ++ + +
+ + + + + + + ..............
Engineering Mathematics - I
24
SOLUTION
(a)
( )
( ) ( ) ( )2 2 2 2
11 2 3 .... 32
2 11 2 3 ..... 2 116
n
nnnu
nn nn n
++ + + +
= = =++ + + +
+
Take 1
nvn
= ; 3 3 0
2 1 2n
n nn
u nLt Ltv n→∞ →∞
⎛ ⎞= = ≠⎜ ⎟+⎝ ⎠
nu∑ and nv∑ behave alike by comparison test.
But nv∑ is diverges by p-series test. Hence nu∑ is divergent.
(b) ( ) ( )
( )( )
( )2 2 2
23 3 32
2 11 2 2 11 2 .... 6
1 2 ..... 3 114
n
nn n nnu
n n nnn
++ ++ + +
= = =+ + ++
Hint : Take 1
nvn
= and proceed as in (a) and show that nu∑ is divergent.
Exercise 1.1 1. Test for convergence the infinite series whose thn term is:
(a) 1
n n− [Ans : divergent]
(b) 1n nn
+ − [Ans : convergent]
(c) 2 1n n+ − [Ans : divergent]
(d) 2 1n
n − [Ans : convergent]
(e) 3 31n n+ − [Ans : divergent]
(f) ( )1
1n n + [Ans : divergent]
(g) 2 1n
n + [Ans : convergent]
(h) 3
5
2 54 1nn
++
[Ans : convergent]
Sequences and Series
25
2. Determine whether the following series are convergent or divergent.
(a) 1 2 3
1 2 31 3 1 3 1 3− − −+ + +
+ + + ............ [Ans : divergent]
(b) 3 3 3 3
12 22 32 2 10........ .....1 2 3
nn
++ + + + + ....... [Ans : convergent]
(c) 1 1 1 ......
1 2 2 3 3 4+ + +
+ + + ........ [Ans : divergent]
(d) 2 2 2
2 3 4 ......3 4 5
+ + + ......... [Ans : divergent]
(e) 2 3 4
1 1 11 2 3
+ + + .......... [Ans : convergent]
(f) 3 2
241
14 2 3n
nn n
∞
=
+
+ +∑ ............... [Ans : divergent]
(g) ( )1
1
8 1n∞
−∑ .................... [Ans : divergent]
(h) 3
51
3 85 9
nn
∞ ++∑ ........................ [Ans : convergent]
(i) 1 2 3
1.3 3.5 5.7+ + + ........... [Ans : divergent]
1.3.3 D’ Alembert’s Ratio Test
Let (i) nu∑ be a series of +ve terms and (ii) ( )1 0n
nn
uLt ku
+
→∞= ≥
Then the series nu∑ is (i) convergent if k < 1 and (ii) divergent if k > 1. Proof :
Case (i) ( )1 1n
nn
uLt ku
+
→∞= <
From the definition of a limit, it follows that
0m∃ > and ( ) 10 1 n
n
ul l l n mu
+< < ∋ < ∀ ≥
Engineering Mathematics - I
26
i.e., 1m
m
u lu
+ < , 2
1
m
m
u lu
+
+
< ,..........
∴ 1 2 ......m m mu u u+ ++ + + ........ = 1 21 .....m mm
m m
u uuu u
+ +⎡ ⎤+ + +⎢ ⎥
⎣ ⎦
1 2 1
11
m m m
mm m m
u u uu . .....
u u u
( ) ( )2 11 ... . 11m mu l l u l
l< + + + = <
−
But 1.
1mul−
is a finite quantity ∴ nn m
u∞
=∑ is convergent
By adding a finite number of terms 1 2 1...... mu u u −+ + + , the convergence of the
series is unaltered. nn m
u∞
=∑ is convergent.
Case (ii) 1 1n
nn
uLt ku
+
→∞= >
There may be some finite number of terms in the beginning which do not satisfy
the condition 1 1n
n
uu
+ ≥ . In such a case we can find a number ‘m’
1 1,n
n
u n mu
+∋ ≥ ∀ ≥
Omitting the first ‘m’ terms, if we write the series as 1 2 3u u u+ + + ........., we
have
32 4
1 2 3
1, 1, 1uu uu u u
≥ ≥ ≥ .......... and so on
∴ 32 21 2 1
1 2 1
...... 1 . .....nuu uu u u u
u u u⎛ ⎞
+ + + = + + +⎜ ⎟⎝ ⎠
(to n terms)
≥ u1 (1 + 1 + 1.1 +.......to n terms) = 1nu
11
.n
nn nn
Lt u Lt n u→∞ →∞
=
≥∑ which → ∞ ; ∴ nu∑ is divergent .
Sequences and Series
27
Note: 1 The ratio test fails when k = 1. As an example, consider the series, 1
1p
n n
∞
=∑
Here 1 1 111 1
pp
n
n n nn
u nLt Lt Ltu n n
+
→∞ →∞ →∞
⎛ ⎞⎛ ⎞ ⎜ ⎟= = =⎜ ⎟+ ⎜ ⎟+⎝ ⎠ ⎝ ⎠
i.e., k = 1 for all values of p, But the series is convergent if p > 1 and divergent if 1p ≤ , which shows that
when k = 1, the series may converge or diverge and hence the test fails .
Note: 2 Ratio test can also be stated as follows:
If nu∑ is series of +ve terms and if 1
n
nn
uLt ku→∞
+
= , then nu∑ is convergent
If k > 1 and divergent if k < 1 (the test fails when k = 1).
Solved Examples
Test for convergence of Series
EXAMPLE 28
(a) 2 3
1.2 2.3 3.4x x x
+ + + ................. (JNTU 2003)
SOLUTION
( );
1=
+
n
nxu
n n
( )( )1
1 1 2
+
+ =+ +
n
nxu
n n;
( )( )( )1
1 1 1.21 2 1
nn
nn
n nu x xu n n x
n
++ +
= =+ + ⎛ ⎞+⎜ ⎟
⎝ ⎠
.
Therefore 1n
nn
uLt xu
+
→∞=
∴ By ratio test nu∑ is convergent When |x| < 1 and divergent when | x | > 1;
When x = 1, ( )2
11 1nu
n n=
+; Take 2
1nv
n= ; 1n
nn
uLtv→∞
=
∴ By comparison test nu∑ is convergent.
Hence nu∑ is convergent when 1x ≤ and divergent when 1x > .
Engineering Mathematics - I
28
(b) 2 31 3 5 7 .......x x x+ + + +
SOLUTION
( ) 12 1 ;−= − nnu n x ( )1 2 1+ = + n
nu n x ; 1 2 12 1
n
n nn
u nLt Lt x xu n
+
→∞ →∞
+⎛ ⎞= =⎜ ⎟−⎝ ⎠
∴ By ratio test nu∑ is convergent when 1x < and divergent when 1x >
When 1: 2 1;nx u n= = − nnLt u→∞
= ∞ ; ∴ nu∑ is divergent.
Hence nu∑ is convergent when 1x < and divergent when 1x ≥
(c) 21 1
n
n
xn
∞
= +∑ ...........
SOLUTION
2 1
n
nxu
n=
+ ;
( )
1
1 21 1
n
nxu
n
+
+ =+ +
.
Hence 2
12
12 2
n
n
u n xu n n
+ ⎛ ⎞+= ⎜ ⎟+ +⎝ ⎠
, ( )
( )2
21
22
11
2 21
n
n nn
nu nLt Lt x xu n
n n
+
→∞ →∞
⎡ ⎤+⎢ ⎥
⎢ ⎥= =⎛ ⎞⎢ ⎥+ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
∴ By ratio test, nu∑ is convergent when 1x < and divergent when 1x > When
2
11:1nx u
n= =
+ ; Take 2
1nv
n=
∴ By comparison test, nu∑ is convergent when 1x ≤ and divergent when 1x >
EXAMPLE 29
Test the series 2
2
1 , 01
n
n
n x xn
∞
→∞
⎛ ⎞−>⎜ ⎟+⎝ ⎠
∑ for convergence.
SOLUTION
( )( )
221
1 22
1 11 ;1 1 1
n nn n
nnu x u xn n
++
⎡ ⎤+ −⎛ ⎞−= = ⎢ ⎥⎜ ⎟+ + +⎢ ⎥⎝ ⎠ ⎣ ⎦
Sequences and Series
29
( )( )
221
2 2
12 .2 2 1
n
n nn
nu n nLt Lt xu n n n
+
→∞ →∞
⎡ ⎤+⎛ ⎞+⎢ ⎥= ⎜ ⎟+ + −⎢ ⎥⎝ ⎠⎣ ⎦
( )( )
( )( )4 2
4 2 2
1 2 1 1
1 2 2 1 –1n
n n nLt x
n n n n→∞
⎡ ⎤+ +⎢ ⎥= =
+ +⎢ ⎥⎣ ⎦
∴ By ratio test, nu∑ is convergent when 1x < and divergent when 1x > when x = 1,
2
2
11n
nun
−=
+ Take 0
1nv
n=
Applying p-series and comparison test, it can be seen that nu∑ is divergent when x = 1.
∴ nu∑ is convergent when x < 1 and divergent 1x ≥
EXAMPLE 30
Show that the series2 3 41 .....2 3 4
p p p
+ + + + , is convergent for all values of p.
SOLUTION
p
nnun
= ; ( )
1
11
p
n
nu
n+
+=
+
( )1 1
1
pn
pn nn
nu nLt Ltu n n
+
→∞ →∞
⎡ ⎤+= ×⎢ ⎥
+⎢ ⎥⎣ ⎦=
( )1 1
1
p
n
nLtn n→∞
⎧ ⎫+⎪ ⎪⎛ ⎞⎨ ⎬⎜ ⎟+ ⎝ ⎠⎪ ⎪⎩ ⎭
= ( )
1 11 0 11
p
n nLt Lt
n n→∞ →∞
⎛ ⎞× + = <⎜ ⎟+ ⎝ ⎠;
nu∑ is convergent for all ‘ p ‘ .
EXAMPLE 31
Test the convergence of the following series
1 1 1 1
1 3 5 7p p p p+ + + + ............
SOLUTION
( )
1 ;2 1
=−
n pun
( )1
12 1
+ =+
n pun
Engineering Mathematics - I
30
( )( )
( )( )
1 2 1 2 . 1 1 22 1 2 1 1 2
p pp pn
p pp pn
n n nuu n n n
+ − −= =
+ +; 1 1n
nn
uLtu
+
→∞=
∴ Ratio test fails.
Take 1
n pvn
= ; ( )
12 1 12 1
2
pn
p pn p
u nv n
n
= =− ⎛ ⎞−⎜ ⎟
⎝ ⎠
; 1 ,2
npn
n
uLtv→∞
=
which is non – zero and finite ∴ By comparison test, nu∑ and nv∑ both converge or both diverge.
But by p – series test, nv∑ = 1
pn∑ converges when p > 1 and diverges
when 1p ≤ ∴ nu∑ is convergent if p > 1 and divergent if 1p ≤ .
EXAMPLE 32
Test the convergence of the series ( ) 0;1
13 >
+∑∞
=
xn
xn
n
n
SOLUTION
( ) ( )
( )
1
1 33
1 2;
1
n n
n n
n x n xu u
n n
+
+
+ +=
+
( ) ( )
3311
32 2. . .
1 1 11nn
nn
u n n n nx xu n x n nn
++ + +⎛ ⎞⎛ ⎞= = ⎜ ⎟⎜ ⎟+ + +⎝ ⎠⎝ ⎠+
13
21 1 .1 11 1
n
n nn
u nLt Lt x xu
n n
+
→∞ →∞
⎛ ⎞+⎜ ⎟= =⎜ ⎟
⎛ ⎞⎜ ⎟+ +⎜ ⎟⎝ ⎠ ⎝ ⎠
∴ By ratio test, nu∑ converges when x < 1 and diverges when x > 1 .
When x = 1, 3
1n
nun+
=
Take 2
1nv
n= ; By comparison test nu∑ is convergent ( give proof )
∴ nu∑ is convergent if 1x ≤ and divergent if x > 1.
Sequences and Series
31
EXAMPLE 33 Test the convergence of the series (JNTU 2002)
(i) 2
21
12n
n
nn
∞
−
⎛ ⎞+⎜ ⎟
⎝ ⎠∑ (ii)
2.5.8 2.5.8.1111.5.9 1.5.9.13
+ + + ... (iii) 1 1.2 1.2.33 3.5 3.5.7
+ + +
SOLUTION
(i) 2
21
12n
n
nn
∞
−
⎛ ⎞+⎜ ⎟
⎝ ⎠∑ =
2
21 1
12n
n n
nn
∞ ∞
= =
+∑ ∑ Let 2
2
1;2n nn
nu vn
= =
( ) ( )2 2
11 1 1 2
1 1 2; .2 2
nn
n n nn
n nuuu n
++ + +
+ += =
21 1 1 1. 1 1
2 2n
n nn
uLt Ltu n
+
→∞ →∞
⎛ ⎞= + = <⎜ ⎟⎝ ⎠
∴ By ratio test nu∑ is convergent. By p –series test, nv∑ is convergent.
∴ The given series ( )n nu v+∑ ∑ is convergent.
(ii) Neglecting the first term, the series can be taken as, 2.5.8 2.5.8.111.5.9 1.5.9.13
+ +
Here, 1st term has 3 fractions ,2nd term has 4 fractions and so on .
∴ thn term contains ( n + 2 ) fractions 2. 5. 8.......are in A. P. ∴ ( )2 thn + term = 2 + ( n + 1 ) 3 = 3n + 5 ;
∴ 1. 5. 9,.......are in A. P. ∴ ( )2 thn + term = 1 + ( n + 1 ) 4 = 4n + 5
∴ ( )( )
2.5.8..... 3 51.5.9..... 4 5n
nu
n+
=+
( )( )( )( )1
2.5.8..... 3 5 3 81.5.9..... 4 5 4 9n
n nu
n n+
+ +=
+ +
( )( )
1 3 84 9
n
n
nuu n
+ +=
+ ; 1
833 1
9 44
n
n nn
nu nLt Ltu n
n
+
→∞ →∞
⎛ ⎞+⎜ ⎟⎝ ⎠= = <⎛ ⎞+⎜ ⎟⎝ ⎠
∴ By ratio test, nu∑ is convergent.
Engineering Mathematics - I
32
(iii) 1, 2, 3, ........ are in A. P thn term = n ; 3. 5. 7..........are in A.P. thn term = 2n + 1
∴ ( )
1.2.3.....3.5.7..... 2 1n
nun
⎡ ⎤= ⎢ ⎥+⎣ ⎦
( )
( )( )1
1.2.3..... 13.5.7..... 2 1 2 3n
n nu
n n+
⎡ ⎤+= ⎢ ⎥+ +⎣ ⎦
1 12 3
n
n
u nu n
+ +⎛ ⎞= ⎜ ⎟+⎝ ⎠
1
1. 11 1
3 22
n
n nn
nu nLt Ltu n
n
+
→∞ →∞
⎛ ⎞+⎜ ⎟⎝ ⎠= = <⎛ ⎞+⎜ ⎟⎝ ⎠
∴ By ratio test, nu∑ is convergent.
EXAMPLE 34
Test for convergence ( ) ( )1
1
1.3.5.... 2 1. 0
2.4.6....2n
n
nx x
n
∞−
=
−>∑ (JNTU 2001)
SOLUTION
The given series of +ve terms has ( ) 11.3.5.... 2 1
.2.4.6....2
nn
nu x
n−−
=
and ( )( )1
1.3.5.... 2 12.4.6.... 2 2
nn
nu x
n+
+=
+
1 2 12 2
n
n nn
u nLt Lt xu n
+
→∞ →∞
+⎛ ⎞= ⎜ ⎟+⎝ ⎠ =
( )( )
12 1 2 .22 1 2
n
n nLt x xn n
→∞
+=
+
∴ By ratio test, nu∑ is converges when x < 1 and diverges when x > 1 when x = 1, the
test fails.
Then ( )1.3.5.... 2 1
12.4.6.....2n
nu
n−
= < and 0nnLt u→∞
≠
∴ nu∑ is divergent. Hence nu∑ is convergent when x < 1, and divergent when 1x ≥
Sequences and Series
33
EXAMPLE 35
Test for the convergence of ( )2 12 6 2 21 ........ ..... 05 9 2 1
nn
nx x x x−⎛ ⎞−+ + + + + >⎜ ⎟+⎝ ⎠
(JNTU 2003) SOLUTION
Omitting 1st term, ( )12 2 , 22 1
nn
n nu x n−⎛ ⎞−= ≥⎜ ⎟+⎝ ⎠
and ' 'nu are all +ve.
( )( )
1
1 1
2 2
2 1
nn
n nu x
+
+ +
−=
+;
11
1
2 2 2 1. .2 1 2 2
n nn
n nn nn
uLt Lt xu
++
+→∞ →∞
⎛ ⎞ ⎛ ⎞ ⎛ ⎞− += ×⎜ ⎟ ⎜ ⎟ ⎜ ⎟+ −⎝ ⎠ ⎝ ⎠⎝ ⎠
= ( )
( )( )( )
1
11
1 12 1 2 12 2. .1 22 1 2 12 2
n nn n
n n nn n
Lt x x+
→∞ ++
⎡ ⎤− +⎢ ⎥ =⎢ ⎥+ −⎢ ⎥⎣ ⎦
;
Hence, by ratio test, nu∑ converges if x < 1 and diverges if x > 1.
When x = 1, the test fails. Then 2 2 ; 1 02 1
n
n nn nu Lt u
→∞
−= = ≠
+; ∴ nu∑ diverges
Hence nu∑ is convergent when x < 1 and divergent x > 1
EXAMPLE 36
Using ratio test show that the series ( )
0
3 4!
n
n
in
∞
=
−∑ converges (JNTU 2000)
SOLUTION
( )3 4!
n
niu n
−= ; ( ) 1
13 4
( 1)!
n
niu n
+
+−= + ; 1 3 4 0 1
1n
n nn
u iLt Ltu n
+
→∞ →∞
⎛ ⎞ −⎛ ⎞= = <⎜ ⎟ ⎜ ⎟+⎝ ⎠⎝ ⎠
Hence, by ratio test, nu∑ converges.
EXAMPLE 37
Discuss the nature of the series, ( )2 32 3 4 ........ 03.4 4.5 5.6
x x x x+ + + ∞ > (JNTU 2003)
SOLUTION Since x > 0 , the series is of +ve terms ;
Engineering Mathematics - I
34
( )( )( )
( )( )( )
11
1 22 3 3 4
n nn n
n nu x u x
n n n n+
+
+ += > =
+ + + +
( )( )( )
21 2 .
1 4n
nn
n xuLtu n n
+
→∞
⎡ ⎤+= ⎢ ⎥
+ +⎢ ⎥⎣ ⎦ =
( )
( )22 2
22
1 .;
5 41n
n
n xLt x
n n n→∞
⎡ ⎤+⎢ ⎥ =⎢ ⎥
+ +⎢ ⎥⎣ ⎦
Therefore by ratio test, nu∑ converges if x < 1 and diverges if x >1
When x = 1, the test fails; Then ( )
( )( )1
2 3n
nu
n n+
=+ +
;
Taking 1
nvn
= ; 1 0n
nn
uLtv→∞
= = ≠
∴ By comparison test nu∑ and nv∑ behave same way. But nv∑ is divergent by p-
series test. (p = 1); ∴ nu∑ is diverges when x = 1
∴ nu∑ is convergent when x < 1 and divergent when 1x ≥
EXAMPLE 38
Discuss the nature of the series ( )( )
3.6.9.....3 .54.7.10..... 3 1 3 2
nnn n+ +∑ (JNTU 2003)
SOLUTION
Here, ( ) ( )
3.6.9.....3 5 ;4.7.10..... 3 1 3 2
n
nnu
n n=
+ +
( )
( )( )( )
1
1
3.6.9.....3 3 3 5;
4.7.10..... 3 1 3 4 3 5
n
n
n nu
n n n
+
+
+=
+ + +
( )( )( )( )
1 3 2 3 3 .53 4 3 5
n
n nn
n nuLt Ltu n n
+
→∞ →∞
+ +=
+ +
= ( )( )
( )( )2
2
325.9 1 13 3 5 1549 1 13 3
n
n n nLtn n n
→∞
⎡ ⎤+ +⎢ ⎥ = >⎢ ⎥+ +⎣ ⎦
∴ By ratio test, nu∑ is divergent.
Sequences and Series
35
EXAMPLE 39
Test for convergence the series 1
1
n
nn
∞−
=∑
SOLUTION 1 n
nu n −= ; ( )1 1 nnu n −
+ = + ;
( ) n
n
n
n
n
n
n
nn
nnnn
nn
uu
⎟⎠⎞
⎜⎝⎛
+=
+=
+=+
11
)1(1–1
–1
1 1 1 1. 0. 0 111
n
n
n nn
uLt Ltu n en
+
→∞ →∞
⎛ ⎞⎜ ⎟= = = <⎜ ⎟+⎝ ⎠
∴ By ratio test nu∑ , is convergent
EXAMPLE 40
Test the series3
1
2n
nn
∞
=∑ , for convergence.
SOLUTION
32
nnun
= ; ( )3
1
2 11n
nu
n+
+=
+
( ) ( ) ( )2
3 21
3 3
112 1 11 2
n
n
n nu n nu n n n n
+++ +
= × = =+
;
1 0 1n
nn
uLtu
+
→∞= < ;
∴ By ratio test, nu∑ is convergent.
EXAMPLE 41
Test convergence of the series 2 !n
n
nn∑
SOLUTION
2 !n
n n
nun
= ; ( )
( )
1
1 1
2 1 !1
n
n n
nu
n
+
+ +
+=
+;
Engineering Mathematics - I
36
( )
( )
11
1
2 1 !. 22 ! 11
nn nn
n nn
nu n nu n nn
++
+
+ ⎛ ⎞= = ⎜ ⎟+⎝ ⎠+
( )
1 1 22 111
nnn n
n
uLt Ltu e
n
+
→∞ →∞= = <
+ (since 2 < e < 3)
∴ By ratio test, nu∑ is convergent.
EXAMPLE 42 Test the convergence of the series nu∑ where nu is
(a) 2 1
3 1n
n ++
(b) ( )
( )1
, 02 1
n
ax an
−
>+
(c) 21.2.3....
4.7.10.....3 3nn
⎛ ⎞⎜ ⎟+⎝ ⎠
(d) 1 21 3
n
n
+
+ (e)
3 2
9
3 75 11
nn n xn
⎛ ⎞+⎜ ⎟+⎝ ⎠
SOLUTION
(a) ( )2
11 2
1 1 3 13 1 1
nn
nn nn
nuLt Ltu n
++→∞ →∞
⎡ ⎤+ +⎛ ⎞ += ×⎢ ⎥⎜ ⎟ + +⎢ ⎥⎝ ⎠ ⎣ ⎦
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎠⎞⎜
⎝⎛ +
⎟⎠⎞⎜
⎝⎛ +
⎟⎠⎞⎜
⎝⎛ +
⎟⎠⎞⎜
⎝⎛ ++
∞→=
++
11
22
22
3113
3113
.11
221
nn
nn
nn
nnn
nLt
= 1 13
<
∴ By ratio test, nu∑ is convergent.
(b) ( )
( )11
2 12 3
ann
a nn nn
nu xLt Ltu xn
+−→∞ →∞
⎡ ⎤+⎛ ⎞= ×⎢ ⎥⎜ ⎟
+⎢ ⎥⎝ ⎠ ⎣ ⎦
= ( )( )
12 1 2 .32 1 2
aa a
an a a
n nLt x xn n
→∞
⎡ ⎤+⎢ ⎥ =⎢ ⎥+⎢ ⎥⎣ ⎦
By ratio test, nu∑ convergence if x < 1 and diverges if x > 1.
Sequences and Series
37
When x = 1, the test fails; Then, ( )
12 1
n aun
=+
; Taking 1
n avn
= we have,
2 1
an
n nn
u nLt Ltv n→∞ →∞
⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟+⎝ ⎠⎝ ⎠=
( )1 1 0
212a an
Lt
n→∞
= ≠+
and finite ( since 0a > ).
∴ By comparison test, nu∑ and nv∑ have same property
But p –series test, we have
(i) nv∑ convergent when a > 1
and (ii) divergent when 1a ≤
∴ To sum up, (i) x < 1, nu∑ is convergent a∀ .
(ii) x > 1, nu∑ is divergent a∀ .
(iii) x = 1, a > 1, nu∑ is convergent, and
(iv) x = 1, 1a ≤ , nu∑ is divergent.
(c) ( )
( )( )( )
2
1 1.2.3.... 1 4.7.10.... 3 34.7.10.... 3 3 3 6 1.2.3....
n
n nn
n n nuLt Ltu n n n
+
→∞ →∞
⎡ ⎤+ += ×⎢ ⎥+ +⎣ ⎦
( )( )
21 1 1
3 2 9n
nLt
n→∞
⎡ ⎤+= = <⎢ ⎥
+⎢ ⎥⎣ ⎦ ;
∴ By ratio test, nu∑ is convergent
(d) ( )( )
( )( )
121
11
1 2 1 3
1 3 1 2
n nn
n nn nn
uLt Ltu
+
++→∞ →∞
⎡ ⎤+ +⎢ ⎥= ×
+ +⎢ ⎥⎣ ⎦
=
12
111 2
11
1 12 1 3 122 3 1
1 1 33 1 2 13 2
n nn n
n n nn n
Lt
++
→∞ ++
⎡ ⎤⎛ ⎞ ⎛ ⎞+ +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎛ ⎞⎝ ⎠ ⎝ ⎠⎢ ⎥× = <⎜ ⎟⎛ ⎞ ⎛ ⎞ ⎝ ⎠⎢ ⎥+ +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
∴ By ratio test, nu∑ is convergent.
Engineering Mathematics - I
38
(e) ( ) ( )
( )
3 2 91
9 3
3 1 7 1 5 113 75 1 11
n
n nn
n nu nLt Lt xu nn
+
→∞ →∞
⎡ ⎤+ + + += × ×⎢ ⎥
++ +⎢ ⎥⎣ ⎦
= ( ) ( )
( )( )( )
3 2 93 29
9 393
111 1 5 13 1 7 1 571 3 15 1 11 3
n
nn nn n nLt xnn n n
→∞
⎡ ⎤++ + +⎢ ⎥× ×⎢ ⎥++ +⎢ ⎥⎣ ⎦
= ( ) ( )
( )( )( )
3 23 9
9
9 3939
71 1 113 1 1 5 13 511 71 3 15 1 35
n
n nn nn nLt x xnn nn n
→∞
⎡ ⎤⎧ ⎫+ + +⎨ ⎬ +⎢ ⎥⎩ ⎭⎢ ⎥× × =⎧ ⎫⎢ ⎥++ +⎨ ⎬⎢ ⎥⎩ ⎭⎣ ⎦
∴ By ratio test, nu∑ converges when x < 1 and diverges when x > 1.
When x = 1, the test fails,
Then ( )
( )( )
( )3
699 9
7 73 1 133 3511 115 1 15 5
n
n n nunn n n
+ += =
+ +
Taking 6
1nv
n= , we observe that
3 05
n
nn
uLtv→∞
= ≠
∴ By comparison test and p series test, we conclude that nu∑ is convergent.
∴ nu∑ is convergent when 1x ≤ and divergent when x > 1.
Exercise – 1.2 1. Test the convergency or divergency of the series whose general term is :
(a) nx
n ............................... [Ans : 1 , 1x cgt x dgt< ≥ ]
(b) 1nnx − ........................... [Ans : 1 , 1x cgt x dgt< ≥ ]
(c) 12 22 1
nn
n x −⎛ ⎞−⎜ ⎟+⎝ ⎠
.............. [Ans : 1 , 1x cgt x dgt< ≥ ]
(d) 2
2
11
nn xn
⎛ ⎞+⎜ ⎟−⎝ ⎠
................... [Ans : 1 , 1x cgt x dgt< ≥ ]
(e) n
nn
........................ [Ans: cgt.]
Sequences and Series
39
(f) 4 .n
n
nn
.......................... [Ans: dgt.]
(g) ( )( )
3 1
3 1
n
n
n +
+ ....................... [Ans: cgt.]
2. Determine whether the following series are convergent or divergent :
(a) 2 3
1.2 3.4 5.6x x x
+ + + .............. [Ans : 1 , 1x cgt x dgt≤ > ]
(b) 2 3
2 2 212 3 4x x x
+ + + + .............. [Ans : 1 , 1x cgt x dgt≤ > ]
(c) 21
1.2.3 4.5.6 7.8.9x x
+ + + ...... [Ans : 1 , 1x cgt x dgt≤ > ]
(d) 2 3
21 ..... ....2 5 10 1
nx x x xn
+ + + + ++
[Ans : 1 , 1x cgt x dgt≤ > ]
(e) 2 3
1.2 2.3 3.4x x x
+ + + ............... [Ans : 1 , 1x cgt x dgt> ≤ ]
1.3.4 Raabe’s Test
Let nu∑ be series of +ve terms and let 1
1n
nn
uLt n ku→∞
+
⎧ ⎫⎛ ⎞⎪ ⎪− =⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
Then (i) If k > 1, nu∑ is convergent. (ii) If k < 1, nu∑ is divergent. (The test fails if k = 1)
Proof : Consider the series 1
n pvn
=∑ ∑
1
1 11 1 1 1p p
n
n
v nn n nv n n+
⎡ ⎤ ⎡ ⎤⎡ ⎤ +⎛ ⎞ ⎛ ⎞− = − = + −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
=( )
2
1 11 . .... 12
p ppnn n
⎡ ⎤−⎛ ⎞+ + + −⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦
= ( )1 1.
2p p
pn
−+ + ..........
1
1n
nn
vLt n pv→∞
+
⎧ ⎫− =⎨ ⎬
⎩ ⎭
Engineering Mathematics - I
40
Case (i) In this case, 1
1 1n
nn
uLt n ku→∞
+
⎧ ⎫− = >⎨ ⎬
⎩ ⎭
We choose a number ‘p’ 1k p∋ > > ; Comparing the series nu∑ with
nv∑ which is convergent , we get that nu∑ will converge if after some
fixed number of terms
1 1
1 pn n
n n
u v nu v n+ +
+⎛ ⎞> = ⎜ ⎟⎝ ⎠
i.e. If, ( )
1
1 11 .2
n
n
p pun pu n+
−⎛ ⎞− > + +⎜ ⎟ ∠⎝ ⎠
.........from (1)
i.e., If 1
1n
nn
uLt n pu→∞
+
⎛ ⎞− >⎜ ⎟
⎝ ⎠
i.e., If k > p, which is true . Hence nu∑ is convergent .The second case also
can be proved similarly.
Solved Examples EXAMPLE 43 Test for convergence the series
3 5 71 1.3 1.3.5. . . .....
2 3 2.4 5 2.4.6 7x x xx + + + + (JNTU 2006, 2008)
SOLUTION Neglecting the first tem ,the series can be taken as ,
3 5 71 1.3 1.3.5. . . .....
2 3 2.4 5 2.4.6 7x x x
+ + +
1.3.5....are in A.P. thn term = ( )1 1 2 2 1n n+ − = −
2.4.6...are in A.p. thn term = ( )2 1 2 2n n+ − =
3.5.7.....are in A.P thn term = ( )3 1 2 2 1n n+ − = +
∴ nu ( thn term of the series) ( )
( )2 11.3.5.... 2 1
.2.4.6.... 2 2 1
nn xn n
+−=
+
Sequences and Series
41
( )( )
( )( )2 3
1
1.3.5.... 2 1 2 1.
2.4.6.... 2 2 2 2 3
n
n
n n xun n n
+
+
− +=
+ +
( )( ) ( ) ( )
( )2 31
2 1
1.3.5.... 2 1 2 12.4.6....2. . .2.4.6.... 2 2 2 3 1.3.5.... 2 1
nn
nn
n nu x nu n n n x
++
+
+ +=
+ + −
= ( )
( )( )
2 22 12 2 2 3
n xn n
++ +
∴
22
2 21
2
14 12
2 34 1 12 2
n
n nn
nu nLt Lt x xu n
n n
+
→∞ →∞
⎛ ⎞+⎜ ⎟⎝ ⎠= =
⎛ ⎞⎛ ⎞+ +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∴ By ratio test, nu∑ converges if 1x < and diverges if 1x >
If 1x = the test fails.
Then 2 1x = and ( )( )
( )21
2 2 2 32 1
n
n
n nuu n+
+ +=
+
( )( )
( ) ( )2 21
2 2 2 3 6 51 12 1 2 1
n
n
n nu nu n n+
+ + +− = − =
+ +
2
21
6 514 4 1
n
n nn
u n nLt n Ltu n n→∞ →∞
+
⎧ ⎫⎛ ⎞ ⎛ ⎞+⎪ ⎪− =⎨ ⎬⎜ ⎟ ⎜ ⎟+ +⎪ ⎪ ⎝ ⎠⎝ ⎠⎩ ⎭
2
22
563 1
4 1 24n
nnLt
nn n
→∞
⎛ ⎞+⎜ ⎟⎝ ⎠= = >
⎛ ⎞+ +⎜ ⎟⎝ ⎠
By Raabe’s test, nu∑ converges. Hence the given series is convergent when 1x ≤ an
divergent when 1x > .
EXAMPLE 44 Test for the convergence of the series (JNTU 2007)
2 33 3.6 3.6.91 .....; 07 7.10 7.10.13
x x x x+ + + + >
Engineering Mathematics - I
42
SOLUTION
Neglecting the first term,
3.6.9....3 .
7.10.13....3 4n
nnu xn
=+
( )
( )( )1
1
3.6.9....3 3 3.
7.10.13.... 3 4 3 7n
n
n nu x
n n+
+
+=
+ +
1 3 3 .3 7
n
n
u n xu n
+ +=
+ ; 1n
nn
uLt xu
+
→∞=
∴ By ratio test, nu∑ is convergent when x < 1 and divergent when x > 1.
When x = 1 The ratio test fails. Then
1 1
3 7 4; 13 3 3 3
n n
n n
u unu n u n+ +
+= − =
+ +
1
4 41 13 3 3
n
n nn
u nLt n Ltu n→∞ →∞
+
⎧ ⎫⎛ ⎞⎪ ⎪ ⎛ ⎞− = = >⎨ ⎬⎜ ⎟ ⎜ ⎟+⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭
∴ By Raabe’s test, nu∑ is convergent .Hence the given series converges if 1x ≤ and
diverges if x > 1.
EXAMPLE 45
Examine the convergence of the series ( )
( )
22 2 2
22 2 21
1 .5 .9 .... 4 34 .8 .12 .... 4n
nn
∞
=
−∑
SOLUTION
( )
( )
22 2 2
22 2 2
1 .5 .9 .... 4 34 .8 .12 .... 4n
nu
n−
= ; ( ) ( )
( ) ( )
2 22 2 2
1 2 22 2 2
1 .5 .9 .... 4 3 4 14 .8 .12 .... 4 4 4n
n nu
n n+
− +=
+
( )( )
21
2
4 11
4 4n
n nn
nuLt Ltu n
+
→∞ →∞
+= =
+ (verify)
∴ The ratio test fails. Hence by Raabe’s test, nu∑ is convergent. (give proof)
Sequences and Series
43
EXAMPLE 46
Find the nature of the series ( ) ( )
2
, 02
nnx x
n>∑ (JNTU 2003)
SOLUTION
( )2
.2
nn
nu x
n= ;
( )21
1
1.
2 2n
n
nu x
n+
+
+=
+
( )
( )( )
21 1
2 1 2 2n
n
nu xu n n
+ +=
+ +;
( )
( )( )
22
12
11.
1 2 44 1 12 2
n
n nn
nu xnLt Lt xu n n n
+
→∞ →∞
+= =
+ +
∴ By ratio test, nu∑ converges when 14
<x , i. e ; x < 4; and diverges when x >4;
When x = 4, the test fails.
( )( )
( )21
2 1 2 24 1
n
n
n nuu n+
+ +=
+
( ) ( )2
1
2 2 112 14 1
n
n
u nu nn+
− − −− = =
++;
1
11 12
n
nn
uLt nu→∞
+
⎡ ⎤⎛ ⎞ −− = <⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
∴ By ratio test, nu∑ is divergent
Hence nu∑ is convergent when x < 4 and divergent when x > 4
EXAMPLE 47
Test for convergence of the series ( )4.7.... 3 1
1.2.3....nn
xn+
∑ (JNTU 1996)
SOLUTION
( )4.7.... 3 1
1.2.3....n
n
nu x
n+
= ; ( )( )
( )1
1
4.7.... 3 1 3 41.2.3.... 1 .
nn
n nu x
n n+
+
+ +=
+
( )( )
1 3 4. 3
1n
n nn
nuLt Lt x xu n
+
→∞ →∞
⎡ ⎤+= =⎢ ⎥+⎣ ⎦
Engineering Mathematics - I
44
∴ By ratio test nu∑ converges if 3 1x < i.e., 13
x < and diverges if 13
x > ;
If 13
x = , the test fails
When 13
x = , ( )⎥⎦⎤
⎢⎣⎡
++
=⎥⎦
⎤⎢⎣
⎡
+
1–43311–
1 nnn
uu
nn
n = 1 1
43 4 3n
nn
−⎡ ⎤ = −⎢ ⎥+ ⎛ ⎞⎣ ⎦ +⎜ ⎟⎝ ⎠
1
11 13
n
nn
uLt nu→∞
+
⎡ ⎤− = − <⎢ ⎥
⎣ ⎦
∴ By Raabe’s test, nu∑ is divergent.
∴ nu∑ is convergent when 13
x < and divergent when 13
x ≥
EXAMPLE 48
Test for convergence ( )2 33 4 52 ........... 0
2 3 4x x x x+ + + + > (JNTU 2003)
SOLUTION
The thn term ( ) 11 n
n
nu x
n−+
= ; ( )( )1
21
nn
nu x
n+
+=
+;
( )( )
12
2.
1n
n
n nu xu n
+ +=
+
( )( )
2
12
2
21.
11n
n nn
nu nLt Lt x xu n n
+
→∞ →∞
+= =
+
∴ By ratio test, nu∑ is convergent if x < 1 and divergent if x > 1
If x = 1, the test fails.
Then ( )( )
2
1
11 1
2n
n nn
nuLt n Lt nu n n→∞ →∞
+
⎡ ⎤+⎡ ⎤− = −⎢ ⎥⎢ ⎥ +⎢ ⎥⎣ ⎦ ⎣ ⎦
= ( )
1 0 12n
Lt nn n→∞
⎡ ⎤= <⎢ ⎥+⎣ ⎦
∴ By Raabe’s test nu∑ is divergent
∴ nu∑ is convergent when x < 1 and divergent when 1x ≥
EXAMPLE 49
Find the nature of the series 3 3.6 3.6.9 ......4 4.7 4.7.10
+ + + ∞ (JNTU 2003)
Sequences and Series
45
SOLUTION
( )
3.6.9.....34.7.10..... 3 1n
nun
=+
; ( )
( )( )1
3.6.9.....3 3 34.7.10..... 3 1 3 4n
n nu
n n+
+=
+ +
1 3 33 4
n
n
u nu n
+ +=
+;
( )( )
133 1 3 143 1 3
n
n nn
nu nLt Ltu n n
+
→∞ →∞
+= =
+
Ratio test fails.
∴ 1
3 41 13 3
n
n nn
u nLt n Lt nu n→∞ →∞
+
⎡ ⎤⎧ ⎫ ⎡ + ⎤⎛ ⎞− = −⎨ ⎬⎢ ⎥ ⎜ ⎟⎢ ⎥+⎝ ⎠⎣ ⎦⎩ ⎭⎣ ⎦
( ) ( )
1 113 1 33 1n n
n nLt Ltn n n
→∞ →∞= = = <
+ +
∴ By Raabe’s test nu∑ is divergent.
EXAMPLE 50 If p, q > 0 and the series
( )( )
( )( )( )( )
1.3. 1 1 21 1.3.51 ....2 2.4. 1 2.4.6 1 2
p p p p ppq q q q q q
+ + ++ + + +
+ + +
is convergent , find the relation to be satisfied by p and q.
SOLUTION
( ) ( ) ( )
( ) ( )1.3.5..... 2 1 1 ..... 1
2.4.6.....2 1 ..... 1n
n p p p nu
n q q q n− + + −
=+ + −
[neglecting 1st term]
( )( )
( )( ) ( )( )( ) ( )( )1
1.3.5..... 2 1 2 1 1 ..... 12.4.6.....2 2 2 1 ..... 1n
n n p p p n p nu
n n q q q n q n+
− + + + − +=
+ + + − +
( )( )
( )( )
1 2 12 2
n
n
n p nuu n q n
+ + +=
+ +;
( )( )
( )( )
11 12 1 2 . 112 1 12
n
n nn
pnn nu nLt Ltqu n nn n
+
→∞ →∞
⎡ ⎤++⎢ ⎥= =⎢ ⎥
+ +⎢ ⎥⎣ ⎦
∴ ratio test fails. Let us apply Raabe’s test
Engineering Mathematics - I
46
( )( )( )( )1
2 21 1
2 1n
n nn
q n nuLt n Lt nu p n n→∞ →∞
+
⎡ ⎤⎡ ⎤ ⎧ ⎫+ +⎛ ⎞ ⎪ ⎪− = −⎢ ⎥⎨ ⎬⎢ ⎥⎜ ⎟ + +⎪ ⎪⎢ ⎥⎝ ⎠ ⎩ ⎭⎣ ⎦ ⎣ ⎦
( ) ( )
( )( )2
2 1 2 111 2n
q n p n nLt n
pn n n→∞
⎡ ⎤⎧ ⎫⎢ ⎥+ − + +⎪ ⎪
⎨ ⎬⎢ ⎥⎪ ⎪⎢ + + ⎥⎩ ⎭⎣ ⎦
( ) ( )1 12 1 2 1 2 2 1
2 2n
q p q pn nLt→∞
⎡ ⎤+ − + + − +⎢ ⎥ =⎢ ⎥⎣ ⎦
Since nu∑ is convergent, by Raabe’s test, 2 2 1 1
2q p− +
>
12q p⇒ − > , is the required relation.
Exercise 1.3
1. Test whether the series 1
nu∞
∑ is convergent or divergent where
( )
( )( )
22 2 222 .4 .6 ..... 2 2
.3.4.5..... 2 1 2
nn
nu x
n n−
=−
[Ans : 1 , 1x cgt x dgt≤ > ]
2. Test for the convergence the series
( )
1
4.7.10..... 3 1 nnx
n
∞ +∑ [Ans :
1 1,3 3
x cgt x dgt< ≥ ]
3. Test for the convergence the series :
(i) 2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2
2 .4 2 .4 .5 .7 2 .4 .5 .7 .8 .10 ...3 .3 3 .3 .6 .6 3 .3 .6 .6 .9 .9
+ + + [Ans : divergent]
(ii) ( )2 33.4 4.5 5.6 ..... 01.2 2.3 3.4
x x x x+ + + > [Ans : cgt if x ≤ 1 dgt if , x > 1]
(iii) ( )
( ) ( )1.3.5..... 2 1. 0
2.4.6.....2 2 2
nn x xn n−
>+∑ [Ans : cgt if x ≤ 1 dgt if , x > 1]
Sequences and Series
47
(iv) ( ) ( ) ( ) ( )
2 2 22 31 2 31 ...... 0
2 4 6x x
x x+ + + + >
[Ans : cgt if x 4< and dgt if , x 4≥ ] 1.3.5 Cauchy’s Root Test
Let nu∑ be a series of +ve terms and let1
nnn
Lt u l→∞
= . Then nu∑ is convergent when
l < 1 and divergent when l > 1
Proof : (i) 1
1nnn
Lt u l a→∞
= < ⇒ ∃ +ve number ( )1
' ' 1 ,nnl u n mλ λ λ< < ∋ < ∀ >
(or) ,nnu n mλ< ∀ >
Since 1, nλ λ< ∑ is a geometric series with common ratio < 1 and
therefore convergent. Hence nu∑ is convergent.
(ii) 1
1nnn
Lt u l→∞
= >
∴ By the definition of a limit we can find a number 1
1nnr u n r∋ > ∀ >
i.e., nu n r> ∀ >
i.e., after the 1st ‘r ‘ terms , each term is > 1. nn
Lt u→∞
= ∞∑ ∴ nu∑ is divergent.
Note : When ( )11,n
nn
uLt→∞
= the root test can’t decide the nature of nu∑ . The fact of
this statement can be observed by the following two examples.
1. Consider the series 31
13 13
1 11 : 1n
nn
nn n nun n n
Lt Lt Lt→∞ →∞ →∞
⎛ ⎞⎛ ⎞− = = =⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠
∑
2. Consider the series 1 ,n∑ in which 1
11 1n
nnn n
un
Lt Lt→∞ →∞
= =
In both the examples given above, 1
1nn
nuLt
→∞
= . But series (1) is convergent
(p-series test) And series (2) is divergent. Hence when the limit=1, the test fails.
Engineering Mathematics - I
48
Solved Examples EXAMPLE 51 Test for convergence the infinite series whose thn terms are:
(i) 2
1nn
(ii) 1
(log )nn (iii) 2
1
11n
n⎡ ⎤+⎢ ⎥⎣ ⎦
(JNTU 1996, 1998, 2001)
SOLUTION
(i) 1
2 2
1 1, nn nnu u
n n= = ;
1
2
1 0 1;nnn n
Lt u Ltn→∞ →∞
= = <
By root test nu∑ is convergent.
(ii) 11 1;
(log ) logn
n nnu un n
= = ; 1 1 0 1;
logn
nn nLt u Lt
n→∞ →∞= = <
∴ By root test , nu∑ is convergent.
(iii) 2
11 1;11 11
nn n nn
u u
nn
= =⎛ ⎞⎛ ⎞ ++ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
1 1 1 1;
11→∞ →∞
= = <⎛ ⎞+⎜ ⎟⎝ ⎠
nn n
n nu
en
Lt Lt
∴ By root test nu∑ is convergent.
EXAMPLE 52 Find whether the following series are convergent or divergent.
(i) 1
13 1n
n
∞
= −∑ (ii) 2 3 4
1 1 11 .....2 3 4
+ + + + (iii) ( )
11
1n
nn
n xn
∞
+=
+⎡ ⎤⎣ ⎦∑
SOLUTION
(i)
1
11 1 1
13 1 3 13
n
nn
n nn
n
u
⎛ ⎞⎜ ⎟⎛ ⎞ ⎜ ⎟= =⎜ ⎟− ⎛ ⎞⎜ ⎟⎝ ⎠ −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
Sequences and Series
49
1
1 1 1 11 33 13
n
nnn n n
n
Lt u Lt→∞ →∞
⎛ ⎞⎜ ⎟⎜ ⎟= = <
⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
; By root test, nu∑ is convergent.
(ii) 1
11 1; 0 1n
nn nn nn n
u Lt u Ltn n→∞ →∞
⎛ ⎞= = = <⎜ ⎟⎝ ⎠
; By root test, nu∑ is convergent.
(iii) ( )
1
1n
n n
n xu
n +
+⎡ ⎤⎣ ⎦=
( ){ }
1
1
1
1n n
nn nn n
n xLt u Lt
n +→∞ →∞
⎡ ⎤+⎢ ⎥=⎢ ⎥⎣ ⎦
( )
1
1 1.n n
n
n xLt
n n→∞
⎡ ⎤+⎧ ⎫⎢ ⎥⎨ ⎬⎢ ⎥⎩ ⎭⎣ ⎦
= 11 1.
n n
nLt xn n→∞
+⎛ ⎞⎜ ⎟⎝ ⎠
11 11 .
n nLt x
n n→∞
⎛ ⎞+⎜ ⎟⎝ ⎠
= 11.
n nLt x x
n→∞= 1
1since . 1n nLt x
n→∞
⎛ ⎞=⎜ ⎟
⎝ ⎠
∴ nu∑ is convergent if 1x < and divergent if 1x > and when 1x = the test fails.
Then ( )
1
1 n
n n
nu
n +
+= ; Take
1nv
n=
( ) ( )
1
1 1 1. 1n n n
nn n
n
n nu nv n n n+
+ + ⎛ ⎞= = = +⎜ ⎟⎝ ⎠
; 1n
nn
uLt ev→∞
= >
∴ By comparison test, nu∑ is divergent.
( nv∑ diverges by p –series test )
Hence nu∑ is convergent if 1x < and divergent 1x ≥
EXAMPLE 53
If ( )
2
2
1n
n
nnu
n=
+, show that nu∑ is convergent.
Engineering Mathematics - I
50
( )
2
2
1
1
1n
nn
nnn n
nLt u Ltn→∞ →∞
⎡ ⎤⎢ ⎥=⎢ ⎥+⎣ ⎦
; = ( ) 11
n
nn
n n
n nLt Ltnn→∞ →∞
⎛ ⎞= = ⎜ ⎟+⎝ ⎠+
= 1 1 111
n
nLt
en
→∞
⎛ ⎞⎜ ⎟
= <⎜ ⎟⎜ ⎟+⎝ ⎠
; ∴ nu∑ converges by root test .
EXAMPLE 54
Establish the convergence of the series 2 31 2 3
3 5 7⎛ ⎞ ⎛ ⎞+ + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
........
SOLUTION
2 1
n
nnu
n⎛ ⎞= ⎜ ⎟+⎝ ⎠
.........(verify); 1 1 1
2 1 2n
nn n
nLt u Ltn→∞ →∞
⎛ ⎞= = <⎜ ⎟+⎝ ⎠
By root test, nu∑ is convergent.
EXAMPLE 55
Test for the convergence of 1
.1
n
n
n xn
∞
= +∑
SOLUTION
12
1 .11
nnu x
n
⎛ ⎞⎜ ⎟
= ⎜ ⎟⎜ ⎟+⎝ ⎠
;
12
1 1 .11n
nn nLt u Lt x x
n→∞ →∞
⎛ ⎞⎜ ⎟
= =⎜ ⎟⎜ ⎟+⎝ ⎠
∴ By root test, nu∑ is convergent if 1x < and divergent if 1x > .
When 1x = : 1n
nun
=+
, taking 0
1nv
n= and applying comparison test , it can be
seen that is divergent
nu∑ is convergent if 1x < and divergent if 1x ≥ .
EXAMPLE 56
Show that ( )1
11
nn
nn
∞
=
−∑ converges.
Sequences and Series
51
SOLUTION
( )11
nn
nu n= −
( ) ( )1 1 11 1 1 0 1 since 1n n n
nn n nLt u Lt n Lt n→∞ →∞ →∞
= − = − = < = ;
∴ nu∑ is convergent by root test.
EXAMPLE 57
Examine the convergence of the series whose thn term is 2 .3
nnn x
n+⎛ ⎞
⎜ ⎟+⎝ ⎠
SOLUTION
2 .3
nn
nnu xn
+⎛ ⎞= ⎜ ⎟+⎝ ⎠;
1 23
nnn n
nLt u Lt x xn→∞ →∞
+⎛ ⎞= =⎜ ⎟+⎝ ⎠
∴ By root test, nu∑ converges when 1x < and diverges when 1x > .
When 1x = : 23
n
nnun
+⎛ ⎞= ⎜ ⎟+⎝ ⎠;
21
31
n
n nn n
nLt u Lt
n
→∞ →∞
⎛ ⎞+⎜ ⎟⎝ ⎠=⎛ ⎞+⎜ ⎟⎝ ⎠
= 2
3
1 0ee e
= ≠ and the terms are all +ve .
∴ nu∑ is divergent . Hence nu∑ is convergent if 1x < and divergent if 1x ≥ .
EXAMPLE 58
Show that the series,
1 2 32 3 4
2 3 4
2 2 3 3 4 4 ......1 1 2 2 3 3
− − −⎡ ⎤ ⎡ ⎤ ⎡ ⎤
− + − + − +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
is convergent (JNTU 2002)
( ) 1
1
1 1nn
n n
n nun n
−+
+
⎡ ⎤+ += −⎢ ⎥
⎢ ⎥⎣ ⎦; =
1 1 1nn nn n
n n
−− ⎡ ⎤+ +⎛ ⎞ ⎛ ⎞ −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
1 11 1 1
nn n
n n
−− ⎡ ⎤⎛ ⎞ ⎛ ⎞+ + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
;
111 1 11 1 1
nn
nun n
−− ⎡ ⎤⎛ ⎞ ⎛ ⎞= + + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
Engineering Mathematics - I
52
1 1
1 11 1 1n
n n
=⎛ ⎞ ⎧ ⎫⎪ ⎪⎛ ⎞+⎜ ⎟ + −⎨ ⎬⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭
∴ 1 1 1.
1 1n
nnLt u
e→∞=
−=
1 11e
<−
∴ By root test, nu∑ is convergent.
EXAMPLE 59
Test 1
mm
u∞
=∑ for convergence when
( )2
21
m
m m
eu
m
−
−=
+
SOLUTION
( ) ( )2
1
121
mm
mm mm m
mLt u Lte→∞ →∞
⎡ ⎤+⎢ ⎥= ⎢ ⎥
⎢ ⎥⎣ ⎦
; 21 21 1
m
m
eLt ee m e→∞
⎛ ⎞+ = = >⎜ ⎟⎝ ⎠
Hence Cauchy’s root tells us that mu∑ is divergent.
EXAMPLE 60
Test the convergence of the series 2 .n
ne
∑
SOLUTION
1
10 1
nn
n nn n
nLt u Lte→∞ →∞
= = < ∴ By root test, nu∑ is convergent.
EXAMPLE 61
Test the convergence of the series, ( )2
22 3 1
1 .2 3 ... ......, 01 2
n n
n
n xx x x
n +
++ + + >
SOLUTION
( )
1
1
1
1 .n nnn
n nn n
n xLt u Lt
n +→∞ →∞
⎡ ⎤+= ⎢ ⎥
⎢ ⎥⎣ ⎦ 1
1 1. .n n
nLt xn n→∞
⎡ ⎤+⎛ ⎞= ⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
Sequences and Series
53
11 11 . .
n nLt x
n n→∞
⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
1
1.1. since 1nn
x x Lt n→∞
⎡ ⎤= = =⎢ ⎥⎣ ⎦
∴ By root test, nu∑ converges if 1x < and diverges when 1x > . When 1x = , the test fails.
Then 1 11 .
n
nun n
⎛ ⎞= +⎜ ⎟⎝ ⎠
; Take 1
nvn
=
11 0
nn
n nn
uLt Lt ev n→∞ →∞
⎛ ⎞= + = ≠⎜ ⎟⎝ ⎠
∴ By comparison test and p-series test, nu∑ is divergent.
Hence nu∑ is convergent when 1x < and divergent when 1x ≥ .
Exercise 1.4 1. Test for convergence the infinite series whose thn terms are:
(a) 1
2 1n − ............................. [Ans : convergent]
(b) ( )
( )2
1 . 1log n n ≠ ...................... [Ans : convergent]
(c) 3 1 .4 3
nn xn
+⎛ ⎞⎜ ⎟+⎝ ⎠
................................. [Ans : 4 4,3 3
x cgt x dgt< ≥ ]
(d) nxn
............................................ [Ans : cgt for all 0x ≥ ]
(e) n
nn
............................................. [Ans : convergent]
(f) 3
3 .n nn∠
........................................ [Ans : convergent]
(g) ( )
( )
2
2
2 1
2
n
n
n
n
− ................................... [Ans : convergent]
(h) ( )211
nnn − ................................... [Ans : convergent]
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54
(i) 2
1 nnn
−−⎛ ⎞⎜ ⎟⎝ ⎠
................................... [Ans : divergent]
(j) ( ), 01
nnx xn
⎛ ⎞ >⎜ ⎟+⎝ ⎠ ......................... [Ans : 1x < cgt , 1x ≥ dgt]
2. Examine the following series for convergence :
(a) 2 3
2 31 ....., 02 3 4x x x x+ + + + > .................. [Ans : 1 , 1x cgt x dgt≤ > ]
(b) 2 31 2 3 .....
4 7 10⎛ ⎞ ⎛ ⎞+ + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
....................... [Ans : convergent]
1.3.6 Integral Test +ve term series,
( ) ( ) ( )1 2 ..... .....nφ φ φ+ + + +
where ( )nφ decreases as n increases is convergent or divergent according as the
integral ( )1
x dxφ∞
∫ is finite or infinite.
Proof : Let ( ) ( ) ( )1 2 .....nS nφ φ φ= + + +
From the above figure, it can be seen that the area under the curve y = ( )xφ between any two ordinates lies between the set of exterior and interior rectangles formed by the ordinates at
n = 1, 2, 3, .....n, n +1 ,...... Hence the total area under the curve lies between the sum of areas of all interior rectangles and sum of the areas of all the exterior rectangles.
Hence
( ) ( ) ( ){ } ( ) ( ) ( ) ( ){ }1
11 2 ..... 2 3 ..... 1
nn x dx nφ φ φ φ φ φ φ
++ + + ≥ ≥ + + + +∫
∴ ( ) ( )111n nS x dx Sφ φ
∞
+≥ ≥ −∫
Sequences and Series
55
1 2 3 n1A 2A 3A nA A 1nn +
1B
2B
3B
1nB -
nB1nB +
O1
O2O
3On O 1n +
1P
2P
3P
1c
2c
3c
nc1nc +
OY= ( )n
As ,n → ∞ Lt nS is finite or infinite according as ( )
1x dxφ
∞
∫ is finite or infinite.
Hence the theorem.
Solved Examples EXAMPLE 62
Test for convergence the series 2
1logn n n
∞
=∑ (JNTU 2003)
SOLUTION
2 2
1 1log log
n
ndx Lt dx
x x x x∞
→∞
⎡ ⎤= ⎢ ⎥
⎣ ⎦∫ ∫ = [ ]2
log log n
nLt x→∞
= ∞
∴ By integral test, the given series is divergent.
EXAMPLE 63
Test for convergence of the series 1
1p
n n
∞
=∑ (JNTU 2003)
SOLUTION
1 1
1 1n
p pndx Lt dx
x x∞
→∞
⎡ ⎤= ⎢ ⎥⎣ ⎦∫ ∫ = 1
11
np
n
xLtp
− +
→∞
⎡ ⎤⎢ ⎥− +⎣ ⎦
;
= 11 11
p
nLt n
p−
→∞⎡ ⎤−⎣ ⎦−
Case (i) If p >1, this limit is finite; ∴ 1
pn∑ is convergent.
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56
Case (ii) If p < 1, the limit is in finite; ∴ 1
pn∑ is divergent.
Case (iii) If p = 1, the limit ( )1log logn
n nLt x Lt n→∞ →∞
= = ∞ ; ∴ 1
pn∑ is divergent.
Hence 1
pn∑ is convergent if p > 1 and divergent if p ≤ 1
EXAMPLE 64
Test the series 2
1n
ne
∞
∑ for convergence.
SOLUTION
( )( )2n n
nu n saye
φ= = ;
( )nφ is +ve and decreases as n increases. So let us apply the integral test.
( ) 2
1 1
xx dx xe dxφ∞ ∞
−=∫ ∫ = { }2
1
1 , 22
te dt t x dt xdx∞
− = =∫
= 11 1 1 102 2 2
tee e
− ∞ ⎛ ⎞− = − − =⎜ ⎟⎝ ⎠
, which is finite.
By integral test, nu∑ is convergent.
EXAMPLE 65
Apply integral test to test the convergence of the series 22
1 sinn n
π∞ ⎛ ⎞⎜ ⎟⎝ ⎠
∑
SOLUTION
Let ( ) 2
1 sinnn n
πφ ⎛ ⎞= ⎜ ⎟⎝ ⎠
; ( )nφ decreases as n increases and is +ve .
( ) 22 2
1 sinx dx dxx x
πφ∞ ∞ ⎛ ⎞= ⎜ ⎟
⎝ ⎠∫ ∫ ; Let txπ =
00
22
1 1 1sin costdt t πππ π π
− = =∫ finite, 2 dx dtxπ− = ; 2
1 1dx dtx π= −
∴ By integral test, nu∑ converges 2 2x t π= ⇒ = 0x t= ∞ ⇒ =
Sequences and Series
57
EXAMPLE 66
Apply integral test and determine the convergence of the following series.
(a) 21
34 1
nn
∞
+∑ (b) 3
41
23 2
nn
∞
+∑ (c) 1
13 1
∞
+∑ n
SOLUTION
(a) ( ) 2
34 1
nnn
φ =+
is +ve and decreases as n increases
( ) 21 1
34 1
xx dx dxx
φ∞ ∞
=+∫ ∫
2 14 1 81 5,
x t xdx dt
x t x t
⎛ ⎞+ = ⇒ =⎜ ⎟⎜ ⎟= ⇒ = = ∞ ⇒ = ∞⎝ ⎠
( )1 5
3 3 log t log 58 8
t
n n
dtx dx Lt Ltt
φ∞
→∞ →∞
⎡ ⎤ ⎡ ⎤= = − = ∞⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦∫ ∫
∴ By integral test, nu∑ diverges.
(b) ( )3
4
23 2
nnn
φ =+
decreases as n increases and is +ve
( )3
41 1
23 2
xx dx dxx
φ∞ ∞
=+∫ ∫
= [ ]55
1 1 log6 6
dt tt
∞∞= = ∞∫ [where t 43 2x= + ]
By integral test, nu∑ is divergent.
(c) ( ) 13 1
nn
φ =+
is +ve, and decreases as n increases.
( ) [ ]1 1 4
1 1 13 1 log3 1 3 3 t
dtx dx dx t x tx t
φ∞ ∞ ∞
∞= = = + = = ∞+∫ ∫ ∫
∴ By integral test, nu∑ is divergent.
Alternating Series A series, ( ) 1
1 2 3 4 ..... 1 .....nnu u u u u−− + − + − + + − + , where nu are all +ve, is an
alternating series.
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58
1.3.7 Leibneitz Test
If in an alternating series ( ) 1
11 .n
nn
u∞
−
=
−∑ , where nu are all +ve,
(i) 1, ,n nu u n+> ∀ and (ii) 0nnLt u→∞
= , then the series is convergent.
Proof : Let 1 2 3 4 ......u u u u− + − + be an alternating series (‘ nu ’are all +ve)
Let 1 2 3 4......u u u u> > > , Then the series may be written in each of the following two forms :
( ) ( ) ( )1 2 3 4 5 6 .....u u u u u u− + − + − + .........(A)
( ) ( )1 2 3 4 5 .....u u u u u− − − − − ..........(B) (A) Shows that the sum of any number of terms is +ve and (B) Shows that the sum of any number of terms is 1u< . Hence the sum of the series is finite. ∴ The series is convergent.
Note : If 0nnLt u→∞
≠ , then the series is oscillatory.
Solved Examples
EXAMPLE 67
Consider the series 1 1 11 .....2 3 4
− + −
In this series, each term is numerically less than its preceding term and thn term 0→ as n → ∞ .
∴ By Leibneitz’s test, the series is convergent.
(Note the sum of the above series is 2eLog )
EXAMPLE 68
Test for convergence ( ) 112 1
n
n
−−−∑ (JNTU 1997)
SOLUTION
The given series is an alternating series ( ) 11 nnu−−∑ , where
12 1nu
n=
−
We observe that (i) 0,nu n> ∀ (ii) 1,n nu u n+> ∀ (iii) 0nnLt u→∞
=
∴ By Leibneitz’s test, the given series is convergent.
Sequences and Series
59
EXAMPLE 69
Show that the series 1 1 11 ......3 9 27
S = − + − + converges. (JNTU 2000)
SOLUTION
The given series is ( ) ( )
11
11
11
3
nn
nn u−∞
−
−
−= −∑ ∑ , where 1
13n nu −= is an alternating series
in which 1. 0,nu n> ∀ 2. 1,n nu u n+> ∀ and 3. 0nnLt u→∞
= ;
Hence by Leibneitz’s test, it is convergent.
EXAMPLE 70
Test for convergence of the series, 2 3
2 3 .....,0 11 1 1
x x x xx x x
− + − + < <+ + +
(JNTU 2003) SOLUTION
The given series is of the form( ) ( )
111 .
11
n nn
nn
xu
x
−−−
= −+∑ ∑ ,
where 1
n
n n
xux
=+
Since 0 < x < 1, 0,nu n> ∀ ;
Further, 1
1 11 1
n n
n n n n
x xu ux x
+
+ +− = −+ +
( )( )
( )( )( )
1
1 1
11 1 1 1
nn n
n n n n
x xx xx x x x
+
+ +
−−= =
+ + + + .
0 < x < 1 ⇒ all terms in numerator and denominator of the above expression are +ve. ∴ 1, .n nu u n+> ∀
Again, 0nx → as nx → ∞ since 0 < x < 1; ∴ 0 0
1 0nnLt u→∞
= =+
∴ By Leibneitz’s test, the given series is convergent.
EXAMPLE 71
Test for convergence ( )
( )( )
1
2
1
1 2
n
n n n n
−∞
=
−
+ +∑ (JNTU 2004)
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60
SOLUTION The given series is an alternating series ( ) 11 n
nu−−∑
where ( )( )
11 2
nun n n
=+ +
; 0,nu n> ∀ ;
Again, ( )( )( ) ( )( )1 2 3 1 2n n n n n n+ + + > + +
∴ ( )( )( ) ( )( )
1 1 ,1 2 3 1 2
nn n n n n n
< ∀+ + + + +
i.e., 1 ,n nu u n+ < ∀
Further, ( )( )
1 01 2
nn nLt u Lt
n n n→∞ →∞= =
+ +
∴ By Leibnitz’s test, ( ) 1
21 n
nu∞
−−∑ is convergent
EXAMPLE 72 Test for the convergence of the following series,
1 2 3 4 5 ....6 11 16 21 26
− + − + − + (JNTU 1998, 2004)
SOLUTION
Given series, ( ) ( )1 1
11 1
5 1n n
nn
n un
∞− −
=
− = −+∑ ∑ is an alternating series
05 1n
nu nn
= > ∀+
; ( )( ) 1
1 1 ,5 1 5 6 5 1 5 6 n n
n n u u nn n n n +
+ −− = ⇒ < ∀
+ + + +
Again, 1 0
5 1 5nn n
nLt u Ltn→∞ →∞
= = ≠+
Thus conditions (ii) or (iii) of Leibnitz’s test are not satisfied. The given series is not convergent. It is oscillatory.
EXAMPLE 73 Test the nature of the following series.
(a) ( ) 1
1
11
n
n n
−∞ −
+ +∑ (b)
( ) 1
2
11
n
n
−−+∑ (c)
( ) 1
1
11
n
n n
−∞
=
−+∑
Sequences and Series
61
SOLUTION
(a) 1 0
1nu nn n
= > ∀+ +
;
11 1
1 1 2n nu un n n n+− = −
+ + + + +
( ) ( ) ( )( )( )2 2 0
1 1 2 2 1 1 2n n
n n n n n n n n n n+ −
= = >+ + + + + + + + + + + +
∴ By Leibnitz’s test the series converges.
(b) 2
1 0,1nu n
n= > ∀
+;
( ) 122
1 1 ,1 1 1
n nu u nn n
+> ⇒ > ∀+ + +
;
0nnLt u→∞
= ∴ By Leibnitz’s test, given series converges.
(c) 1 0,
1nu nn
= > ∀+
;
11 12 1 ,
2 1 n nn n u u nn n ++ > + ⇒ < ⇒ > ∀
+ +
By Leibnitz’s test, given series converges.
EXAMPLE 74
Test the convergence of the series 1 1 1 ......
5 2 5 3 5 4− + − + (JNTU 2004)
SOLUTION
The series can be written as ( ) 1
1
15 1
n
n n
−∞
=
−
+∑ ;
15 1nu
n=
+
(i) 0nu n> ∀
(ii) 11 15 2 5 1
5 2 5 1 n nn n u u nn n ++ > + ⇒ < ⇒ > ∀
+ +
(iii) 0nnLt u→∞
= ;
By Leibnitz’s test, the given series converges.
EXAMPLE 75 Test for convergence the series, 1 1 11 .....2 4 6− + − + (JNTU 1997)
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62
SOLUTION
The given series can be written as ( )1
2
n
n−
∑ (omitting 1st term)
1 02
nn
> ∀ ; 11 1 ,2 2 2 n nu u n
n n +> ⇒ > ∀+
; 1 02n
Ltn→∞
=
∴ By Leibnitz’s test, ( )1
2
n
n−
∑ is convergent.
EXAMPLE 76
Test for convergence the series, 1 1 11 ......3! 5! 7!
− + − + (JNTU 2004, 2007)
SOLUTION
General term of the series is ( )
( )
112 1 !
n
n
−−−
The series is an alternating series; ( )
1 02 1 !
nn
> ∀−
( ) ( ) 1
1 1 ,2 1 ! 2 1 ! n nu u n N
n n +> ⇒ > ∀ ∈− −
; ( )
1 02 1 !n
Ltn→∞
=−
By Leibnitz’s test, given series is convergent.
1.4 Absolute convergence A series nu∑ is said to be absolutely convergent if the series | |nu∑ is convergent Ex. Consider the series
3 3 3
1 1 11 ......2 3 4nu = − + − + − +∑
3 3 3 31
1 1 1 11 ......2 3 4nu
n
∞
= + + + + =∑ ∑
By p − series test, nu∑ is convergent ( 3 1p = > )
Hence nu∑ is absolutely convergent.
Note: 1. If nu∑ is a series of +ve terms, then nu∑ = | |nu∑ . For such a series, there is no difference between convergence and absolute convergence. Thus a series of +ve terms is convergent as well as absolutely convergent.
Sequences and Series
63
2. An absolutely convergent series is convergent. But the converse need not be true.
Consider 1
1
1 1 1 1( 1) . 1 ......2 3 4
n
n
∞−− = − + − +∑
This series is convergent (1.7.3)
But 1 1 1 11( 1) . 1 .......2 3 4
n
n−− = + + + +∑ is divergent (p-series test).
Thus nu∑ is convergent need not imply that | |nu∑ is convergent (i.e.,
nu∑ is not absolutely convergent).
1.5 Conditional Convergence If the series nu∑ is divergent and nu∑ is convergent, then nu∑ is said to be conditionally convergent. Ex. Consider the Series
1 1 11 .......2 3 4
− + − nu∑ is convergent by Leibnitz’s test. (Ex.1.7.3)
But 1 1 11 .....2 3 4nu = + + +∑ is divergent by p − series test.
∴ nu∑ is conditionally convergent.
1.6 Power Series and Interval of Convergence A series, 2
0 1 2 ..... .....nna a x a x a x+ + + + + where ‘ na ’ are all constants is a power
series in x .
It may converge for some values of x .
1 1 .n n
n nn n
u aLt Lt xu a
+ +
→∞ →∞= (1st term is omitted.) = kx (say) where 1n
nn
aLt ka
+
→∞=
Then, by ratio test, the series converges when 1kx < .
i.e., it converges 1 1, ( 0)x k
k k−⎛ ⎞∀ ∈ ≠⎜ ⎟
⎝ ⎠
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64
The interval 1 1,
k k−⎛ ⎞
⎜ ⎟⎝ ⎠
is known as the interval of convergence of the given power
series.
Solved Examples EXAMPLE 77
Find the interval of convergence of the series 31
n
n
xn
∞
=∑
SOLUTION
1
13 3;( 1)
n n
n nx xu un n
+
+= =+
3
31 1. .11 1
n
n n nn
u nLt Lt x Lt x xu n
n
+
→∞ →∞ →∞
⎛ ⎞⎜ ⎟⎛ ⎞ ⎛ ⎞= = =⎜ ⎟⎜ ⎟ ⎜ ⎟+⎝ ⎠⎝ ⎠ ⎜ ⎟+⎝ ⎠
By ratio test, the given series converges when 1,x < i.e., ( 1,1)x ∈ −
When 311, ,nx u n= =∑ ∑ which, is convergent by p series test.
∴ nu∑ is convergent when 1x = Hence, the interval of convergence of the given series is (–1, 1)
EXAMPLE 78 Test for the convergence of the following series. (a) 1 1 11 ..........
2 3 4− + − + (JNTU 1996)
(b) 2 2 2 2 2 2 21 1 1 1 1 1 11 .......2 3 4 5 6 7 8+ − − + + − − + (JNTU 1998)
(c) 2 4 6
1 ..........2 4 6x x x− + − + (JNTU 2004)
(d) 0
( 1) ( 1) ,n nn x∞
− +∑ with 12
x < (JNTU 2004)
SOLUTION (a) The series is of the form 1( 1)n
nu−−∑ where 1nu
n=
Sequences and Series
65
It is an alternating series where (i) 0n nu > ∀ (ii) 1n nu u n+> ∀ and (iii) 0;nn
Lt u→∞
= ∴ By Leibnitz test, the series is convergent.
Again the series 1 1 11 ......2 3 4
+ + + + is divergent, by
p − series test. Hence the given series is conditionally convergent.
(b) 21
1n
p
u n
∞
=
=∑ ∑ which is convergent by p − series test .
∴ The given series is absolutely convergent. ∴ It is convergent. (c) The given series is
2 2 2 2
1 1( 1) . . = (-1) ; (2 2)! (2 2)!
n nn n
n nx xu un n
− −− −− ∴ =
− −∑ ∑
2
1 ;2 !
n
nxun+ = 21 1 .
(2 1)(2 )n
n
u xu n n
+ =−
; 1 0 1n
nn
uLtu
+
→∞= <
By ratio test, the series nu∑ converges ;x∀ i.e., nu∑ is absolutely
convergent ;x∀ (d) Here, 1
1( 1) ;| | ( 2)n nn nu n x u n x +
+= + = + (neglect 1st term)
12(1 )( 2) 11( 1) (1 )
n
n n nn
u n nLt Lt x Lt x xu n n
+
→∞ →∞ →∞
++= = = <
+ + 1( )2x <∵
∴ nu∑ is convergent x∀ , i.e., given series is absolutely convergent and hence convergent.
EXAMPLE 79 Show that the series
2 21 .....2 3
x xx+ + + + converges absolutely x∀
SOLUTION
1 0 1n
nn
u xLt
u n+
→∞= = < when 0x ≠ [since
1
1;( 1)! !
n n
n n
x xu u
n n
−
+= =−
]
∴ By ratio test, nu∑ is convergent 0x∀ ≠ .
Engineering Mathematics - I
66
When 0x = , the series is (1 + 0 + 0 + ......) and is convergent
∴ nu∑ converges nu⇒ ∑ is absolutely convergent x∀ .
EXAMPLE 80
Show that the series, 2 4
1 1 11 ........3 3 3
− + − + is absolutely convergent.
SOLUTION
11
13nn
nu
∞
−=
=∑ ∑ , which is a geometric series with common ratio 1 13
<
∴ It is convergent. Hence given series is absolutely convergent.
EXAMPLE 81 Test for convergence, absolute convergence and conditional convergence of the series,
1 1 11 ..........5 9 13
− + − + (JNTU 2003)
SOLUTION
The given alternating series is of the form 1( 1) ,nnu−−∑ where,
14 3nu
n=
−.
Hence, 0 ;nu n N> ∀ ∈ ( )1
1 14 1 3 4 1nu
n n+ = =+ − +
11 1
4 3 4 1n nu un n+− = −
− +
4 1 4 3
(4 3)(4 1)n nn n
+ − +=
− +
4 0,(4 3)(4 1)
n Nn n
= > ∀ ∈− +
i.e., 1,n nu u n N+> ∀ ∈ 1 0;
4 3nn nLt u Lt
n→∞ →∞= =
−
All conditions of Leibnitz’s test are satisfied.
Hence 1( 1)nnu−−∑ is convergent.
1 ;
4 3nun
=−
Take 1
nvn
= ; 1 03 4(4 )
n
n nn
u nLt Ltv n n
→∞ →∞= = ≠
− and finite.
∴ By comparison test, nu∑ and nv∑ behave alike.
Sequences and Series
67
But by p − series test, nv∑ is divergent (since 1)p = .
nu∑ is divergent and ∴ The given series is conditionally convergent.
EXAMPLE 82
Test the series 1
1
1( 1) .3
n
n n
∞−
=
−∑ , for absolute / conditional convergence .
SOLUTION The given series is an alternating series of the form ( ) 11 n
nu−−∑ .
Here
(i) 1 ,
3nu n Nn
= ∀ ∈
(ii) ( )3 1 3 3 1 3 ,n n n n n+ > ⇒ + > ∀ .
∴ 1 1
3 1 3n n<
+ , i.e., 1 ,n nu u n N+ < ∀ ∈
And 1 0
3nn nLt u Lt
n→∞ →∞= =
∴ By Leibnitz’s test, the given series is convergent.
But 1 1 1( 1) .3 3
n
n n−− =∑ ∑ is divergent by p − series test (since
p = 1 12
< )
∴ The given series is conditionally convergent.
EXAMPLE 83
Test the following series for absolute / conditional convergence.
(a) 12
1
sin( )( 1) ,n
n
nn
α∞−
=
−∑ (b) 2
13
1( 1) .
1n
n
nn
∞−
=
−+∑
(c) 0
( 1)(2 )!
n
n n
∞
=
−∑ (d) 13 1( 1) .
nn
n
ne
π−+−∑
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68
SOLUTION
(a) 2 2
sin 1n
nu
n nα
= < since sin 1nα⎡ < ⎤⎣ ⎦ considering 21
nv n= and using
comparison and p - series tests, we get that nu∑ is convergent nu∑ is absolutely convergent .
(b) By Leibnitz’s test, the series converges.
Taking 1
nvn
= , by comparison and p - series tests , 2
3 1n
n +∑ , is seen to
be divergent. Hence given series is conditionally convergent.
(c) Take 1
2 !nun
= ; 1 0 1n
nn
uLt
u+
→∞= < ; By ratio test, nu∑ is convergent;
Hence given series is absolutely convergent.
(d) 3 1
n
n n
nue
π+= ; By root test, is convergent, ∴ given series is absolutely
convergent. [In problems (a) to (d) above, hints only are given. Students are advised to do the complete problem themselves]
EXAMPLE 84 Find the interval of convergence of the following series.
(a) 13
1( 1)
nn
n
xn
∞−
=
−∑ (b) 1
1
( 2)( 1)3
nn
nn
n x∞−
=
+−∑ (c) log(1 )x+
SOLUTION
(a) Let the given series be nu∑ ; Then 3
n
n
xu
n= ;
1
1 3( 1)
n
n
xu
n
+
+ =+
3
1 .1
n
n nn
u nLt Lt xu n
+
→∞ →∞
⎛ ⎞= ⎜ ⎟+⎝ ⎠ =
3
1 .11nLt x x
n→∞
⎛ ⎞⎜ ⎟
=⎜ ⎟⎜ ⎟+⎝ ⎠
∴ By ratio test, nu∑ is convergent if 1x <
i.e., nu∑ is absolutely convergent if 1x < ;
Sequences and Series
69
∴ nu∑ is convergent if 1x <
If x = 1, the given series becomes 3 3 3
1 1 11 ........2 3 4
− + − +
which is convergent, since 3
1n∑ is convergent.
Similarly, if x = –1, the series becomes 3 3
1 1n n
− = −∑ ∑ which is also
convergent.
Hence the interval of convergence of nu∑ is ( )1 1x− ≤ ≤
(b) Proceeding as in (a),
1 23
n
nn
u xLt
u+
→∞
+=
∴ nu∑ is convergent if 2 3x + < , i.e., if 3 2 3x− < + < , i.e., if
5 1x− < < .
If x = –5, ( )2 11 .nnu n−= −∑ ∑ , and is divergent ( in both these cases
If x = 1, ( ) 11 .nnu n−= −∑ ∑ , and is divergent 0nn
Lt u→∞
≠ )
Hence the interval of convergence of the series is (–5 < x < 1)
(c) log (1 + x ) = 2 3 4
.......2 3 4x x xx − + − +
= ( ) 1
11
nn
nn
x un
∞−
=
− =∑ ∑ (say)
n
n
xu
n= ;
1
1 1
n
n
xu
n
+
+ =+
Engineering Mathematics - I
70
1 1 | |11
n
n nn
uLt Lt x x
un
+
→∞ →∞= =
⎛ ⎞+⎜ ⎟⎝ ⎠
By ratio test, nu∑ is convergent when 1x <
i.e., nu∑ is absolutely convergent and hence convergent when –1 < x < 1.
When x = –1, ( ) 1 1 1 1 11 . 1 .......2 3 4
nnu
n−= − = − + − +∑ ∑ ,
which is convergent by Leibnitz’s test. (give the proof )
When x = 1, nu∑ = 1 1 11 ......2 3 4
⎛ ⎞− + + + +⎜ ⎟⎝ ⎠
which is divergent,
since 1n∑ is divergent by p –series test ( prove ) .
Hence nu∑ is convergent when 1 1x− < ≤ .
Interval of convergence is ( 1 1x− < ≤ ).
Exercise 1.5
1. Use integral test and determine the convergence or divergence of the following
series:
1 2
1n∑ ...................................... [Ans : convergent]
2. ( )2
2
1logn n
∞
∑ ....................................... [Ans : convergent]
2. Test for convergence of the following series:
1. 1 1 11 ....2 4 6
− + − + .................................. [Ans : convergent]
2. ( )
( )( )
1
1
12 1 2
n
n n
−∞ −−∑ ................................... [Ans : convergent]
Sequences and Series
71
3. ( )51 2
11 n n
∞−−−∑ ................................... [Ans : convergent]
3. Classify the following series into absolutely convergent and conditionally
convergent series :
1. ( )
3
1 n
n−
∑ ....................................... [Ans : abs.cgt]
2. 32
sin n
n∑ .......................................... [Ans : abs.cgt]
3. ( )( )2
1log
n
n n−
∑ ......................................... [Ans : abs.cgt]
4. Find the interval of convergence of the following series :
1. 2n nx
n∑ ............................................ [Ans : x−∞ < < ∞ ]
2. 2
nxn∑ ............................................ [Ans : 1 1x− ≤ ≤ ]
3. 2 3 4
......2 3 4
x x xx − + − + .................................. [Ans : 1 1x− < ≤ ]
5. (a) Show that 2 2 2
1 1 11 ....2 3 4
− + − + is absolutely convergent.
(b) Show that 1 1 11 ....2 3 4
− + − + is conditionally convergent.
Summary
1. The geometric series 1
1
n
nx
∞−
=∑ converges if | | 1x < , diverges if 1x ≥ , and oscilates
when 1x ≤ − 2. If nu∑ is convergent, 0nn
Lt u→∞
= [The convergent need not be necessary ]
3. p – series test :- 1
1p
n n
∞
=∑ is convergent if p > 1 and divergent if 1p ≤
Engineering Mathematics - I
72
4. Comparison test :- The series nu∑ and nv∑ are both convergent or both
divergent if n
nn
uLtv→∞
is finite and non – zero.
5. D’Alembert’sRatio test :- nu∑ converges or diverges according as
1 1 or 1n
nn
uLtu
+
→∞< >
1
Alternately, if 1 or 1n
nn
uLtu→∞
+
⎛ ⎞> <⎜ ⎟
⎝ ⎠ . If the limit = 1, the test fails
6. Raabe’s test : nu∑ converges or diverges according as
1
1 1 or 1 n
nn
uLt nu→∞
+
⎡ ⎤⎧ ⎫− > <⎨ ⎬⎢ ⎥
⎩ ⎭⎣ ⎦.
7. Cauchy’s root test: nu∑ converges or diverges according as 1
1 or 1 nnn
Lt u→∞
⎛ ⎞< >⎜ ⎟
⎝ ⎠ (If limit = 1, the test fails.)
8. Integral test : A series ( )nφ∑ of +ve terms where ( )nφ decreases as n increases
is convergent or divergent according as the integral ( )1
x dxφ∞
∫ is finite or infinite.
9. Alternating series – Leibnitz’s test: An alternating series ( ) 1
1
1 nn
n
u∞
−
=
−∑ convergent
if (i) nuu nn ∀> + ,1 and (ii) 0nnLt u→∞
=
10. Absolute / conditional convergence : (a) nu∑ is absolutely convergent if | |nu∑ is convergent.
(b) nu∑ is conditionally convergent if nu∑ is convergent and | |nu∑ is
divergent.
(c) An absolutely convergent series is convergent, but converse need not be true .
i.e., a convergent series need not be convergent.
Sequences and Series
73
Miscellaneous Exercise - 1.6 1. Examine the convergence of the following series:
1. 1 1 1 ......
1.2 3.4 5.6+ + + ................................... ... [cgt.]
2. 2 2 2
3 3 3
1 2 3 ......1 1 2 1 3 1
+ + ++ + +
.............................. [dgt.]
3. 2 32 2 2 .......
1 2 3+ + + .............................. [dgt.]
4. 2 3
1 2 3 .....1 2 1 2 1 2
+ + ++ + +
................................ [cgt.]
5. ( )2 3
2 3 ..... 01 1 1
x x x xx x x
+ + + >+ + +
........ [cgt. if 1x ≤ dgt. if x > 1]
6. ( )2 33 42 ..... 0
8 27x xx x+ + + > .......................... [cgt. if 1x ≤ dgt. if x > 1]
7. 1 1.3 1.3.51 ....2 2.4 2.4.6
+ + + + ................................. [dgt.]
8. 2 2 2 2 2 2
2 2 2 2 2 2
3 3 .5 3 .5 .7 ......6 6 .8 6 .8 .10
+ + + ............................... [cgt]
9. 3.4 4.5 5.6 ....1.2 2.3 3.4
+ + + ................................ [dig.]
10. ( ) ( ) ( ) ( )
2 2 32 31 2 3
. .... 02 4 6
x x x x+ + + > ............. [cgt. if x <4, dgt. if 4x ≥ ]
11. ( )2 3
2 2 21 ..... 02 3 4x x x x+ + + + > ......................... [cgt.if 1x ≤ , dgt. if x > 1]
12. ( )2 3
2 33 4 5 ..... 04 5 6x x x x⎛ ⎞ ⎛ ⎞+ + + + >⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ............. [cgt if x < 1, dgt. if 1x ≥ ]
13. 2
11n
n⎛ ⎞+⎜ ⎟⎝ ⎠
∑ .......................... [dgt.]
Engineering Mathematics - I
74
14. 3
2
23
n
n∑ ................................................ [cgt.]
15. 2 , 11
na an
<+∑ ................................................ [cgt.]
16. 1 1 11 .....
2.2 3.3 4.4− + − + − + − ....................... [Abs. cgt.]
2. Examine for absolute and conditional convergence of the following series:
1. ( )3
2
313
nn
n−∑ .................................... [Abs. cgt.]
2. ( )1 .
2
n
n
n−∑ ...................................... [Abs. cgt.]
3. 1 1 1 112 3 4 5
− + − + ........................... [Cond. cgt]
4. ( ) ( )2
3
11 n n
n+
−∑ .............................. [Cond.cgt.]
3. Determine the interval of convergence of the following series :
1. 2 3 4
.....2 3 4x x xx + + + + ............................... [ 1 1x− ≤ < ]
2. ( )1
.2
n
n
xn+
∑ ................................ [ 3 1x− < < ]
Solved University Questions (J.N.T.U)
1. Test the convergence of the series:
1 2 3 .........
1.2.3 2.3.4 3.4.5+ + +
Solution Let nu be the thn term of the series;
Then, ( )( ) ( )( )
11 2 1 2nnu
n n n n n= =
+ + + +
Sequences and Series
75
Let 2
1nv
n= ; then,
( )( )2
1 2n
n nn
u nLt Ltv n n→∞ →∞
=+ +
= 1 1
1 21 1nLt
n n→∞
=⎛ ⎞⎛ ⎞+ +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
,
Which is non-zero and finite.
∴ By comparison test, both nu∑ and nv∑ converge or diverge together.
But nv∑ is convergent by p-series test (p > 1) ∴ nu∑ is convergent.
2. Show that every convergent sequence is bounded Solution
Let na be a sequence which converges to a limit ‘l ‘say.
nnLt a l→∞
= ⇒ given any +ve number ∈ , however small ,
we can always find an integer ‘ m ‘ , , na l∋ − <∈ , n m∀ ≥
Taking ∈ = 1, we have, 1na l− < ;
i.e., ( ) ( )1 1nl a l− < < + , n m∀ ≥
Let λ = min ( ){ }1 2 1, ,........ , 1ma a a l− − , and μ = max ( ){ }1 2 1, ,........ , 1ma a a l− +
Then obviously, naλ μ≤ ≤ , n N∀ ∈ ;
Hence na is bounded.
3. Show that the geometric series 2
0
1 .......m
m
q q q∞
=
= + + +∑ converges to the sum
11 q−
when 1q < and diverges when 1q ≥
(JNTU 2001) Solution See theorm 1.2.3 (replace ‘x ‘ by ‘q’ ) .
Engineering Mathematics - I
76
4. Define the convergence of a series. Explain the absolute convergence and conditional convergence of a series. Test the convergence of series
(JNTU 2000)
2
11n
n
−⎡ ⎤
+⎢ ⎥⎣ ⎦∑
Solution For theory part , refer 1.2.1, 1.2.2, 1.8, 1.8.1, 1.9.1 and 1.9.2
Problem : Let
2
11n
nun
−⎛ ⎞= +⎜ ⎟⎝ ⎠
; ( )1 11n
nnn n
Lt u Ltn
−
→∞ →∞
⎛ ⎞= +⎜ ⎟⎝ ⎠
= 2
1 1 111
nnLt
e
n
→∞= <
⎡ ⎤+⎢ ⎥⎣ ⎦
By Cauchy’s root test, nu∑ is convergent.
5. Test the convergence of the series, 2 31 1.3 1.3.51 .....2 2.4 2.4.6
x x x+ + + +
(JNTU 2001) Given that x > 0. Solution Omitting the first term of the series, we have,
( )1.3.5. 2 1
2.4.6....2n
n
nu x
n−
= ; ( )( )
11
1.3.5..... 2 1.
2.4.6..... 2 2n
n
nu x
n+
+
+=
+;
1 2 1 .2 2
n
n nn
u nLt Lt x xu n
+
→∞ →∞
+⎛ ⎞= =⎜ ⎟+⎝ ⎠
By ratio test, nu∑ is convergent when x < 1, and divergent when x > 1
The ratio test fails when x = 1
When x = 1, 1
2 2 11 12 1 2 1
n
n
u nu n n+
+− = − =
+ +
1
11 12 1 2
n
n nn
u nLt n Ltu n→∞ →∞
+
⎡ ⎤⎛ ⎞ ⎛ ⎞− = = <⎢ ⎥⎜ ⎟ ⎜ ⎟+⎝ ⎠⎝ ⎠⎣ ⎦ ;
∴ By Raabe’s test , nu∑ diverges . ∴ The given series converges when x < 1 and diverges when 1x ≥ .
Sequences and Series
77
6. Test the convergence of the series, 2 3
2 31 2 3 4 ......... 02 3 4 5
x x x x⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + + + >⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
(JNTU 2002) Solution Neglecting the 1st term,
12
n
nnu xn
⎡ + ⎤⎛ ⎞= ⎜ ⎟⎢ ⎥+⎝ ⎠⎣ ⎦;
1
11122 1
nn
n nu x xn n
⎛ ⎞++⎛ ⎞ ⎜ ⎟= =⎜ ⎟+ ⎜ ⎟+⎝ ⎠ ⎝ ⎠
1n
nnLt u x→∞
= ; By Cauchy’s root test, nu∑ is cgt. when x < 1 and dgt. when
x > 1; when x = 1 , the test fails .
When x = 1, ( )( )
11
21
n
n nnu
n
+=
+; 2
1 0nn
eLt ue e→∞
= = ≠
∴ nu∑ is divergent.
∴ ∑ nu is cgt. when x < 1 and dgt. when 1x ≥ .
7. Test the series whose thn term is ( )3 1 2nn − for convergence. (JNTU 2003) Solution
( )3 1
2n n
nu
−= ;
( ){ }1 1
3 1 12n n
nu + +
+ −= ;
( )( )
1 3 22 3 1
n
n
nuu n
+ +=
− 1 1 1
2n
nn
uLtu
+
→∞= < ;
∴ By ratio test, nu∑ is convergent.
8. Show by Cauchy’s integral test that the series ( )2
1log p
n n n
∞
=∑ converges if p > 1
and diverges if 0 1p< ≤ (JNTU 2003) Solution
Let ( )( )
1log px
x xφ = ; 2x ≥ ; Then ( )xφ decreases as x increases in [ ]2,∞
Engineering Mathematics - I
78
( )( )2 2 log p
dxx dxx x
φ∞ ∞
=∫ ∫ log 2
p
duu
∞
= ∫ 1
log 21
pup
−∞=
− ;
[Taking log x = u, 1 dx dux
= 2 log 2x u= ⇒ = and x u= ∞ ⇒ = ∞ ]
Case (i) : 1 1 0p p> ⇒ − < ⇒ Integral is finite , and Case (ii) : 0 1p< ≤ ⇒ Integral is infinite.
Hence, by integral test, the given series converges if p > 1 and diverges when 0 1p< ≤ .
9. Test the convergence of the series ∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛+
23
11n
n
Solution
;11
111
12
3
1
n
nn
nn
n
nu
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛+
1 1 1nnn
Lt ue→∞
= < [ 2 < e < 3 ] .
By Cauchy’s root test, nu∑ is convergent.
10. Test the convergence of the series,( )
( )2
1 .1
n n
n
xn n
∞
=
−−∑ , 0 < x < 1
Solution
The given series is of the form ( )1 nnu−∑ , where
( )1
n
nxu
n n=
− .
This is an alternating series in which (i) 0nu > and 1n nu u n N+> ∀ ∈ . Further 0nn
Lt u→∞
= . Hence, by Leibnitz test, the series is convergent.
11. Discuss the convergence of the series, 2 4 61 ..........
2 1 3 2 4 3 5 4x x x
+ + + +
(JNTU 1995, 2002, 2003, 2008) Solution
thn term of the series = ( )
2
2 1
n
nxu
n n=
+ + (omitting 1st term)
Sequences and Series
79
( )
2 2
1 3 2
n
nxu
n n
+
+ =+ +
; ( )
21 2 1 .3
n
n
u n n xu n
+ + +=
+
( )
2 212 11 . 1
.31
n
n nn
u n nLt Lt x xu
n
+
→∞ →∞
⎡ ⎤+ +⎢ ⎥= =⎢ ⎥+⎢ ⎥⎣ ⎦
;
∴ By ratio test, nu∑ converges if 2 1x < , i.e., if 1x < , and diverges if
2 1x > , i.e., if 1x > ;
When 2 1x = , ( )
12 1nu
n n=
+ +; taking 3
2
1nv
n= ,
( )
32
32
12 11 1
n
n nn
u nLt Ltv n n n
→∞ →∞= =
+ +
∴ By comparison test, nu∑ and nv∑ both converge or diverge together;
But nv∑ is convergent by p-series test.
∴ nu∑ is convergent if 1x ≤ and divergent if 1x > .
12. Test the convergence of the series ( )
2
1 1
n
n
xn n
∞
= +∑ (JNTU 2006, 2007)
Solution
( )
2
1
n
nxu
n n=
+;
( )
2 2
1 2 1
n
nxu
n n
+
+ =+ +
( )
2 21111 . .
22 1n
n
u n n nx xu n
n
+++
= =+ +
; 21n
nn
uLt xu
+
→∞= ;
∴ By ratio test, nu∑ converges when 1x < and diverges for 1x > .
When 1x = , ( )3
2
111
nun n
=+
taking nv 32
1
n= and applying the comparision
test, we observe that nu∑ is convergent .
Hence nu∑ converges when 1x ≤ and diverges when 1x > .
Engineering Mathematics - I
80
13. Find the interval of convergence of the series, 2 3 4
.........2 3 4x x x
+ + + ∞
(JNTU 2006, 2007)
Solution
For the given series, 1
1
n
nxun
+
=+
; 2
1 2
n
nxun
+
+ =+
11121
n
n nn
u nLt Lt x xu n
+
→∞ →∞
⎛ ⎞+⎜ ⎟= =⎜ ⎟+⎝ ⎠
By ratio test, nu∑ converges when 1x < i.e., 1 1x− < <
When x = 1, 1
1nun
=+
Taking 1
nun
= ; 111
n
n
uv n
=+
and, 1 0n
nn
uLtv→∞
= ≠ and finite.
∴ Both nu∑ and nv∑ converge or diverge together.
But nv∑ diverges ∴ nu∑ also diverges when x = 1.
When x = –1, the given series is
1 1 1 12 3 4 5
− + − + which is alternating series with
1n nu u n+> ∀ and 0nu → as n → ∞
∴ By Leibneitz’s test nu∑ converges when x = –1
∴ Interval of convergence is [(–1, 1) i.e., 1 1x− ≤ <
14. Test the convergence of the series ( )( )1
1.3.5.... 2 12.5.8.... 3 2n
nn
∞
=
++∑ (JNTU 2006)
Solution
( )( )
1.3.5.... 2 12.5.8.... 3 2n
nu
n+
=+
; ( )( )1
1.3.5.... 2 32.5.8.... 3 5n
nu
n+
+=
+
Sequences and Series
81
1 2 33 5
n
n
u nu n
+ +=
+;
( )( )
132 2 15 33
n
n nn
u nLt Ltu
n
+
→∞ →∞
⎡ ⎤+⎢ ⎥= = <⎢ ⎥+⎣ ⎦
∴ By ratio test, nu∑ is convergent.
15. Prove that the series ( )( )∑
∞
=22
1
n
n
nlogn– converges absolutely. (JNTU 2006)
Solution
( )3
1log
nun n
= ; ( )3 2
2 log 2logdx dt
tx x
∞ ∞
=∫ ∫
(where t = log x ) log 21 1
log 2t∞−
= = , which is finite.
∴ By integral test nu∑ is convergent.
∴ nu∑ converges absolutely.
Note: This problem can also be done by Leibnitz test. The reader is advised to try that method also.
16. Test the convergence of the series ( )
3
2 1, 0
1nn
x xn
+>
+∑ (JNTU 2006)
Solution
thn term of the given series , 3
2 11
nn
nu xn
+=
+;
( )
( )1
1 3
2 1 11 1
nn
nu x
n+
+
⎡ ⎤+ += ⎢ ⎥
+ +⎢ ⎥⎣ ⎦
( )1
3
2 31 1
nn xn
++=
+ +
( )( ){ }
( )( )
1 31
3
12 3 .2 11 1
n
nnn n
n
nn xuLt Ltu x nn
+
+
→∞ →∞
++= ×
++ +
( ) ( )
( ) ( )
33
33
3
3 12 1 . 12 .11 11 .2 1 2
n
n nn nLt x xn nn nn
→∞
⎡ ⎤+ +⎢ ⎥
⎢ ⎥ =⎧ ⎫⎢ ⎥+ + +⎨ ⎬⎢ ⎥⎩ ⎭⎣ ⎦
By ratio test, nu∑ converges if x < 1 and diverges if x > 1. If x = 1 the test fails.
Engineering Mathematics - I
82
When x = 1, 3
2 11n
nun
+=
+; Taking 2
1nv
n= ;
23
2 1 2 01
n
n nn
u nLt Lt nv n→∞ →∞
+= × = ≠
+ and finite
∴ nu∑ and nv∑ converge or diverge together .
But nv∑ converges ∴ nu∑ also converges.
Thus, nu∑ converges when 1x ≤ and diverges when x > 1.
17. Test the series ( ) ( )
21
1 logn
n
nn
∞
=
−∑ , for absolute/conditional convergence
(JNTU 2006) Solution
( ) ( )
2
1 logn
n
nu
n−
= ; ( )
2
logn
nu
n= ;
22 log 2
log tx dx te dtx
∞ ∞−=∫ ∫ [taking log x = t , tx e= , 1 logdx tx = ]
[ ] ( )log 2log 2
10 1 log 2 . log 2 12
t tte e e− − ∞ −= − + = − − = − ,
which is finite.
∴ By integral test nu∑ is convergent nu⇒ ∑ converges absolutely.
(Note that nu∑ is cgt. by Leibnitz’s test).
18. Test the convergence of the series ( )
1log log nn
∑ (JNTU 2006)
Solution
Given that ( )
1log log
n nun
= ;
1 1 0 1
log logn
nn nLt u Lt
n→∞ →∞
⎡ ⎤= = <⎢ ⎥
⎣ ⎦
By Cauchy’s root test, nu∑ is convergent.
Sequences and Series
83
19. Find the interval of convergence of the series,
3 5 71 1 3 1.3.5. . . . ......
2 3 2 4 5 2.4.6 7x x xx + + + + (JNTU 2007)
Solution
Term of the series, ( )
( )2 11.3.5..... 2 1
.2.4.6.....2 2 1
n
n
n xun n
+−=
+ (neglecting 1st term)
( )( )
( ) ( )2 3
1
1.3.5..... 2 1 2 1.
2.4.6.....2 2 2 2 3
n
n
n n xun n n
+
+
− +=
+ + ;
( )
( )( )
221 2 1
.2 2 2 3
n
n nn
nuLt Lt xu n n
+
→∞ →∞
⎡ ⎤+= ⎢ ⎥
+ +⎢ ⎥⎣ ⎦
( )( )
22
2 2
22
4 14.
10 64n
n n nLt x xn n n
→∞
⎡ ⎤+ +⎢ ⎥ =⎢ ⎥+ +⎢ ⎥⎣ ⎦
By ratio test, nu∑ converges when 2 1x < , i.e., 1 1 1x x< ⇒ − < <
When 2 1x = , the test fails;
Then 2
2 21
4 10 6 8 51 14 2 1 4 2 1
n
n
u n n nu n n n n+
⎛ ⎞+ + +− = − =⎜ ⎟+ + + +⎝ ⎠
( )
( )2
21 2
581 2 1
2 14n
n nn
nu nLt n Ltu n n n
→∞ →∞+
+⎡ ⎤⎛ ⎞− = = >⎢ ⎥⎜ ⎟
⎝ ⎠ + +⎣ ⎦
∴ By Raabe’s test, nu∑ converges when 2 1x = , i.e., 1x = ± .
∴ Interval of convergence of nu∑ is [–1 < x < 1]
20. Test the series 2
1 2 3 ... ,2 3 8 –1
nn
+ + + + for convergence or divergence.
[JNTU 2006] Solution
un = 2 1n
n + ; Let νn = 3/2
1n
n
n
uν
= 3/2
2
. 1
n nn +
=2
2 1n
n + =
2
111n
+
Engineering Mathematics - I
84
2
1 = 111+
n
n nn
uLt Lt
nν→∞ →∞
⎛ ⎞⎜ ⎟
= ⎜ ⎟⎜ ⎟⎝ ⎠
which is non-zero finite number
∴ By comparison test, Σun and Σνn behave alike.
But Σνn is convergent by p-series test 3p 12
⎛ ⎞= >⎜ ⎟⎝ ⎠∵
Hence Σun is convergent
21. Test the convergence of the series, 2 2 2
2 1 3 1 4 1 .......3 1 4 1 5 1
− − −+ + +
− − −
[JNTU 2007] Solution
un = 2
1 1 ;( 2) –1
n+n
−+
Let νn = 3/2
1n
[∵ Highest degree of n in Dr – Nr = 2 – 1/2 = 3/2]
n
n
uν
= ( )3/2
2
. 1 1
2) –1
n n+
(n
−
+ =
2
22
1 11
4 31
nn n
nn n
= 1 nnn
n
uLt uν→∞
⇒ Σ and Σνn both converge or diverge together (by comparison
test). But Σνn converges by p-series test 3p 1 .2
⎛ ⎞= >⎜ ⎟⎝ ⎠∵ Hence Σun is
convergent.
22. Test the convergence of 3 3 1n nΣ + − [JNTU 2008]
Solution
un can be written as, 3 3 3 3
3 3
1 1
1n
n n n nu
n n
i.e. un = 3
3
111+ +1nn
⎡ ⎤⎢ ⎥⎣ ⎦
; Let νn = 3/2
1n
Sequences and Series
85
Then, n
n
uν
=
3
111+ +1n
⇒ 1 0.2
n
nn
uLtν→∞
= ≠
∴ By comparison test, Σun and Σνn have same property
Σνn is convergent by p-series test 3p 1 .2
⎛ ⎞= >⎜ ⎟⎝ ⎠∵ Hence Σun is convergent.
23. Test for the convergence of the series, 2 3
1 .... 02 5 10x x x x+ + + + >
[JNTU 1998, 1985, 2002, 2002] Solution
Neglecting the first term, we observe that the nth term of the series,
un = 2 ;1
nxn +
un+1 = 1
2 ,2 2
nxn n
+
+ + so that,
2
12
+1 = + 2 + 2
n
n nn
u nLt Lt x xu n n
+
→∞ →∞
⎛ ⎞= ⎜ ⎟
⎝ ⎠
∴ By ratio test, Σun converges when x < 1 and diverges when x > 1 and the test fails when x = 1
∴ when x = 1, un = 2
11n +
; Taking Σνn = 2
1n
and using comparison test, we can
show that Σun is convergent. [This part of proof is left to the reader as an exercise].
∴ Σun converges for x < 1 and diverges for x > 1.
24. Test the convergence of 2
. 1
nn xn
Σ+
(x > 0) [JNTU 2003]
Solution
2
12
+1 1 . 2 2
n
n nn
u n nLt Lt . xu nn n
+
→∞ →∞
+=
+ +
= 2
2
11+ . 1 1 . 1. 2 21
n
xnLt x x
nn n
→∞+ = =
+ +
Engineering Mathematics - I
86
∴ By ratio test, Σun converges when x < 1 and diverges when x > 1 and when x = 1 the test fails.
When x = 1, un = 2 1n
n +; Taking νn = 1 ,
n 1 n
nn
uLtν→∞
= (verify)
By comparison test, Σun diverges. [Since Σνn diverges by p-series test as p = 1 12
< ]
Hence Σun converges when x < 1 and diverges when x > 1.
25. Test for convergence the series nx
nΣ (x > 0) [JNTU 2007, 2008]
Solution
un = ,nx
n un+1 =
1
1
nxn
+
+ ; 1
1n
n nn
u nLt Lt x xu n
+
→∞ →∞
⎛ ⎞= =⎜ ⎟+⎝ ⎠
∴ By comparison test, Σun converges when x < 1 and diverges when x > 1. when x = 1, the test fails.
When x = 1, un = 1n
and Σun is divergent (p series test – p = 1)
∴ Σun is convergent when x < 1 and divergent when x > 1.
26. Find whether the series 2
1sin n (–1)
(n–1)n
n
∞
=
⎛ ⎞⎜ ⎟⎝ ⎠Σ is absolutely convergent or
conditionally convergent. [JNTU 2006] Solution When n > 2, we have,
nu =
1sin n
(n–1)
⎛ ⎞⎜ ⎟⎝ ⎠ ; Let νn = 3/2
1n
;
∴ n
n
uν
= 3/2
n–1n 1sin
n⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
Sequences and Series
87
1sin = 11 1
n
n nn
u nnLt Ltn
nν→∞ →∞
⎛ ⎞⎜ ⎟⎛ ⎞ ⎛ ⎞⎜ ⎟ =⎜ ⎟ ⎜ ⎟−⎝ ⎠⎜ ⎟⎝ ⎠ ⎜ ⎟⎝ ⎠
∴ By comparison test Σ nu and Σνn behave alike. But Σνn is convergent by
p-series test (p = 3/2 > 1).
∴ Σ nu is convergent. Hence Σun is absolutely convergent.
27. Test whether the series 21
cos n +1n
n π∞
=Σ converges absolutely [JNTU 2006]
Solution
The given series is 21
cos n +1n
n π∞
=Σ =
1( 1) n
nu∞
Σ − where un = 2
1 .1n +
It is obvious that u1 > u2 > …. un > un + 1 > …. and 0nnLt u→∞
=
∴ By Leibnitz’s test given series is convergent.
2
1( 1) . 1
nnu
nΣ − = Σ
+ which is convergent (Take ν = 2
1n
and apply comparison
test. This is an exercise to the reader)
Hence given series is absolutely convergent.
28. Find whether the following series converges absolutely or conditionally
1 1 1 1.3.5 1.3.5.7 . .....6 6 3 6.8.10 6.8.10.12
− + − + [JNTU 2007]
Solution The given series is 16
1 3.5 3.5.71 .....3 8.10 8.10.12
⎡ ⎤− + − +⎢ ⎥⎣ ⎦
= 16
2 3.5 3.5.7 .....3 8.10 8.10.12
⎡ ⎤⎧ ⎫+ − +⎨ ⎬⎢ ⎥⎩ ⎭⎣ ⎦
We can take the series as 3.5 3.5.7 .....,8.10 8.10.12
− + (neglecting other terms) = Σun,
where 3.5.7.....(2 3) ,8.10.12.....(2 8)n
nun+
=+
so that 13.5.7.....(2 3) (2n 5)
8.10.12.....(2 8) (2n 10)nnun+
+ +=
+ +
Engineering Mathematics - I
88
It can be seen that 1
n
nn
uLtu→∞
+
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
= (2 10) 1,(2 5)n
nLtn→∞
⎡ ⎤+=⎢ ⎥+⎣ ⎦
so that ratio test fails.
1
1n
nn
uLt nu→∞
+
⎛ ⎞−⎜ ⎟⎜ ⎟
⎝ ⎠ = (2 10) 1
(2 5)n
nLt nn→∞
⎡ ⎤+−⎢ ⎥+⎣ ⎦
= 5 5 12 5 2n
nLtn→∞
⎛ ⎞ = >⎜ ⎟+⎝ ⎠
∴ By Raabe’s test, nuΣ is convergent. Hence the given series is absolutely
convergent.
29. Test for absolute convergence of the series whose nth term is ( 1) (x 2) .2 5
n
n
− ++
[JNTU 2007]
Solution: Let given series be Σun. Then 22 5n n
xu +=
+
∴ 1 1
2 = ,2 5n n
xu + +
++
= so that, 11
52 12 5 2 = =
52 5 2 22
nn n
nn
nnn
uu
++
⎛ ⎞+⎜ ⎟+ ⎝ ⎠+ ⎛ ⎞+⎜ ⎟
⎝ ⎠
∴ 1 1 1 ;2
n
nn
uLtu
+
→∞= <
∴ By ratio test, nuΣ is convergent. Hence the given series is absolutely
convergent.
30. Test whether the following is absolutely convergent or conditionally
convergent.
( )1
1 ( 1) n 1n n
∞+Σ − + − [JNTU 2008]
Solution
The given series is Σun where un = (– 1)n+1 n 1 .n⎡ ⎤+ −⎣ ⎦ It is an alternating
series.
n 1 n+ − = n 1 n 1 1
n 1 n 1 n
n n
n nν
⎡ ⎤ ⎡ ⎤+ − + +⎣ ⎦ ⎣ ⎦ = =+ + + +
(say),
Sequences and Series
89
(1) νn > 0 ∀ n ; (2)nLt→∞
νn = 0 and (3) νn + 1 > νn ∀ n; since all conditions of
Leibnitz’s test are satisfied, the given series is convergent.
Further, 1 = ;n 1nu
n+ + Taking 1 ,
nnν = we have,
1 = 0211 1
n
n nn
u nLt Ltn
nν→∞ →∞
= ≠⎡ ⎤
+ +⎢ ⎥⎣ ⎦
∴ By comparison test, nuΣ and Σνn behave alike. But Σνn = 1 n
Σ is
divergent by p-series test 1 1 .2
p⎛ ⎞= <⎜ ⎟⎝ ⎠
∴ nuΣ is divergent. Hence the given series is conditionally convergent
31. Find the interval of convergence of the series,
2 3
1 1 1 ........1 2(1 ) 3(1 )x x x
+ + +− − −
[JNTU 2007, 2008]
Solution: If the nth term of the given series is un,
un = 1(1 )nn x−
; un+1 = 1
1( 1) (1 )nn x ++ −
1+1
1 (1 ) = ( +1) (1 – ) 1
nn
nn nn
u n xLt Ltu n x
+
→∞ →∞
⎡ ⎤ −⎢ ⎥⎣ ⎦
= 1 1 1 . = 1 (1 ) 11
nLt
x xn
→∞ − −⎛ ⎞+⎜ ⎟⎝ ⎠
Now, by ratio test:
(i) x = 1 ⇒ the limit is infinite ⇒ Σun is divergent
(ii) x ≠ 0 and x > 1 ⇒ the limit is < 1 ⇒ Σun is convergent
Engineering Mathematics - I
90
(iii) x ≠ 0 and x < 1 and > 0 ⇒ the limit is > 1 ⇒ Σun is divergent
(iv) If x = 0, the series is 1 + 1 1 .....,2 3
+ + which is divergent by p-series test
(p = 1)
(v) If x < 0, the limit is < 1 ⇒ Σun is convergent by ratio test .
Hence the given series converges for all values of (i) x > 1 and x ≠ 0 and (ii) x < 0.
∴ The interval of convergence of the given series is (– ∞ , 0) ∪ (1, ∞ ).
Objective Type Questions
1. The infinite series 21
11n n
∞
= +∑ is
(i) convergent (ii) divergent (iii) oscillatory (iv) none of these [Ans :(i)]
2. The series 2
11
nn
++
is
(i) convergent (ii) divergent
(iii) oscillatory (iv) none of these [Ans :(ii)]
3. The series 1 1 1 1 .....
1 1 2 2 3 3 4 4− + − +
(i) oscillatory (ii) absolutely convergent
(iii) conditionally convergent (iv) none of these [Ans :(ii)]
4. The series 1 1 11 ......2 3 4
− + − is
(i) oscillatory (ii) divergent
(iii) convergent (iv) none of these [Ans :(iii)]
Sequences and Series
91
5. The interval of convergence of the series 2 3 4
......2 3 4x x xx − + − + ,is
(i) x−∞ < < ∞ (ii) 1 2x− < <
(iii) 1 1x− < ≤ (iv) none of these [Ans :(iii)]
6. The series 1 2 3 .........
1.2 3.4 5.6+ + + ∞ is
(i) convergent (ii) divergent
(iii) oscillatory (iv) none of these [Ans :(ii)]
7. The series 1 2 3 .........
1.3 3.5 5.7+ + + ∞ , is
(i) conditionally convergent (ii) convergent
(iii) divergent (iv) none of these [Ans :(iii)]
8. The series 2 3 4 5 ........
1 2 3 4p p p p+ + + + ∞ is convergent if
(i) p < 2 (ii) p = 2
(iii) p > 2 (iv) none of these [Ans :(iii)]
9. The series 6 – 10 + 4 + 6 – 10 + 4 + 6 – 10 + 4 + 6............. ∞ is
(i) convergent (ii) oscillatory
(iii) divergent (iv) none of these [Ans :(ii)]
10. The series 1 1 1 .........
2.4 4.6 6.8+ + + is
(i) convergent (ii) divergent
(iii) oscillatory (iv) none of these [Ans :(i)]
Engineering Mathematics - I
92
2. Indicate whether the following statements are true or false:
1. The series 1
1 2 n−+∑ is convergent . ........................ [False]
2. The series 2
2
52 7nn
++∑ is not convergent . ...................... [True]
3. The series 1 1 1 ........1 2 3
+ + + is divergent.................... [False]
4. The series 3 5
.......,3 5x xx − + − + − converges when 1 1x− ≤ ≤ .. [True]
5. The series ( ) 11
.5
n
nn
−−∑ is absolutely convergent. ........................ [True]
6. The series 2 3 42 3 4 ....x x x x+ + + + ∞ is convergent if x > 1........ [False]
7. The series 2 3
........2 3x xx + + + ∞ is divergent if 1x ≥ ............... [True]
8. The series 3 3 42 3 4
2! 3! 4!1 ......2 3 4 5x x x x+ + + + + + ∞ is convergent
if x > e ........................... [True]
9. The series
32
11n
n
−⎛ ⎞+⎜ ⎟⎝ ⎠
∑ is divergent ............................. [False]
10. The series 2 3
2 31 2 3 4 .......2 3 4 5
x x x⎛ ⎞ ⎛ ⎞+ + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
is convergent
if x < 1 . ............................. [True]
11. The series ( )2 31 2 3 4 ..... 1x x x x− + − + ∞ < is divergent............ [False]
Sequences and Series
93
12. The series 2 3 4
2 3 4 ......1 1 1 1
x x x xx x x x
− + − + ∞+ + + +
is convergent ...... [True]
13. The series 3 3 3
sin sin 2 sin 3 ......1 2 3
x x x− + + ∞ converges absolutely...... [True]
14. The series ( ) 11 n
n
−−∑ is conditionally convergent. ........................ [True]
15. The series whose thn term is ( )
23 52 a
nn
+
+ is convergent. [False]
3. Fill in the Blanks:
1. The geometric series 1
1
n
nar
∞−
=∑ converges if___________ . [Ans: | | 1r < ]
2. If a series of +ve terms nu∑ is convergent, nnLt u→∞
= ______. . [Ans: 0]
3. { }3 3
11
nn n
∞
=
+ −∑ is __________. [Ans: convergent.]
4. If ( )
3
1
3 45 p
n
nn
∞
=
−
+∑ is divergent ,value of p is _________. [Ans: 4≤ ]
5. The interval of convergence of nu∑ where 22
2
22
n
nnu xn
⎛ ⎞−= ⎜ ⎟+⎝ ⎠
, is _______.
[Ans: 1 1x− < < ]
6. nu∑ is convergent series of +ve series. Then ( )1 nnn
Lt u→∞
is_________.
[Ans: 1< ]
Engineering Mathematics - I
94
7. The series 8 – 12 + 4 + 8 – 12 – 4 + .......is _________. [Ans : Oscillatory ]
8. If 0,nu n> ∀ and nu∑ is convergent, then 1
1n
nn
uLt nu→∞
+
⎡ ⎤⎧ ⎫−⎨ ⎬⎢ ⎥
⎩ ⎭⎣ ⎦ is _______.
[Ans: 1> ]
9. If the series ( ) ( )1
1 , 0nn n
n
a a n∞
=
− > ∀∑ is convergent, then for all values of n , 1
n
n
aa +
is______. [Ans: 1> ]
10. If 2
111 ,n
nn nn
u Lt un
−
→∞
⎛ ⎞= + =⎜ ⎟⎝ ⎠
_____________. [Ans: 1 e ]